UC-NRLF 


$B    35    flDD 


r 


^ppuwy— ^^— i»^ 


■SS»' 


ANALYTIC  GEOMETRY 


A  FIRST  COURSE 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://wwvy.archive.org/details/analyticgeometryOOmaltrich 


About  six  years  ago  I  prepared  a  text  on  elementary 
Analytic  Geometry  for  my  sophomores.  Three  years  ago 
i    revised    it.    and    this    fall   i    expect   to    use  the    second 

revision,  a  copy  OF  WHICH  I  ENCLOSE.  THE  AIM  OF  THE  BOOK 
IS  THE  DEVELOPMENT  OF  POWER  ON  THE  PART  OF  THE  STUDENT. 
I  HAVE.  ACCORDINGLY.  ATTEMPTED  TO  GIVE  A  CLEAR  EXPOSITION 
OF  THE  METHODS  OF  ANALYTIC  GEOMETRY,  BUT  HAVE  LEFT  THE 
WORKING     OUT     OF    THEOREMS     LARGELY    TO    THE    STUDENT. 

I  AM  SENDING  THE  BOOK  OUT  IN  ITS  PRESENT  FORM  IN  THE 
HOPE  THAT  SOME  OF  THOSE  WHO  RECEIVE  IT  MAY  BE  SUFFICIENTLY 
INTERESTED  TO  GIVE  ME  THE  BENEFIT  OF  THEIR  CRITICISM  FOR 
THE  NEXT  REVISION.  IF  ANY  ONE  SHOULD  BE  WILLING  TO  GIVE 
THE  BOOK  A  CLASSROOM  TEST  I  CAN  SUPPLY  A  LIMITED  NUMBER 
OF    COPIES    AT    TWENTY-FIVE    CENTS    EACH. 

All  COMMUNICATIONS   SHOULD    BE   ADDRESSED  TO 

W.    H.    MALTBIE. 

WOMAN'S  College  of  Baltimore. 

BALTIMORE.    MD. 


ANALYTIC  GEOMETRY 


A  FIRST  COURSE 


BY 


WILLIAM  y.  MALTBIE 


1906 

THE  SUN  JOB  PRINTING  OFFICE 

BALTIMORE 


COPYRIGHT,  1906 

BY 

WILLIAM   H.   MALTBIE 


-r' 


H^(^ 


TABLE  OF  CONTENTS 


Article.  Page. 

1,  2     Introduction . , 3 

AXALYTIC    GEOMETRY    OF    ONE    DIMENSION. 

3  The  system  of  co-ordinates 5 

4  The  relation  of  the  point  and  the  equation 6 

5  The  equation  with  equal  roots 7 

6  The  equation  with  complex  roots 8 

7  Transformation  of  co-ordinates  in  one  dimension 9 

8  The  derivation  of  the  formula  of  transformation  which  will 

produce  a  given  result 11 

9  Distance  ratios  and  anharmonic  ratios 12 

SOME  FUNDAMENTAL  IDEAS  OF  THE  ANALYTIC  GEOMETRY  OF  TWO  DIMENSIONS, 

10  The  system  of  co-ordinates 14 

11  The  significance  of  two  simultaneous  equations 16 

12  The  significance  of  a  single  equation.     Loci 17 

TO    CONSTRUCT    THE    LOCUS    WHEN    THE    EQUATION    IS    GIVEN. 

13  The  process  of  curve  plotting 19 

14  The  utility  of  the  process 22 

TO    CONSTRUCT    THE    LOCUS    WHEN    A    FINITE    NUMBER    OF    POINTS    UPON    IT  IS 

GIVEN. 

15  The  method  of  plotting 24 

16  Choice  of  scales  of  measurement 25 

17  Determination  of  intermediate  values 26 

18  Definition  and  test  of  continuity 27 

19  Infinite    discontinuity 28 

TO    DEDUCE   THE   EQUATION    WHEN    THE   RESTRICTIONS    ON    THE    MOVEMENT  OF 
THE    TRACING    POINT    ARE    GIVEN. 

20  The  general  method 29 

21  Distance  between  two  points  in  terms  of  their  co-ordinates. ...  30 

22  Co-ordinates  of  a  point  with  a  given  distance  ratio 30 

23  Angle  between  a  line  and  the  X  axis 32 

24  Illustrations  of  the  development  of  equations  of  loci 32 

TO   DEDUCE   THE   EQUATION   WHEN    A   FINITE   NUMBER   OF    POINTS    ON    A   LOCUS 
OF    SOME    KNOWN    TYPE    IS    GIVEN. 

25  Limitations  of  the  problem.     Outline  of  the  method 35 

26  Number  of  points  required  to  determine  a  curve 36 

27  Complex  conditions  equivalent  to  two  or  more  simple  ones. ...  37 


Table  of  Coxtexts. 

TRANSFORMATION    OF    CO-ORDINATES. 

Article.  Page. 

28  The  general  problem 38 

29  Movement  of  the  axes  parallel  to  themselves 38 

30  Rotation  of  the  axes 40 

31  Change  of  the  angle  between  the  axes 42 

32  Change  of  the  scale  of  measurement 43 

33  The  general  transformation. 43 

34  Transformations  interpreted  as  changes  of  the  loci 43 

INTERSECTION    OF   LOCI. 

35  The  significance  of  imaginaries 45 

36  Possibility  of  error 46 

THE  EQUATION  OF   THE  FIRST   DEGREE   AND   THE    STRAIGHT   LINE. 

37  Standard  forms  of  the  equation  of  the  straight  line 48 

38  The  equation  of  the  straight  line  through  two  given  points. ...  51 

39  The  straight  line  given  by  two  equations  in  three  variables 52 

40  Distance  from  a  point  to  a  line. 53 

41  Normal  form  of  the  equation  of  a  straight  line 54 

42  Intersections  of  lines 56 

43  Families  of  lines 57 

44  The  line  at  infinity 58 

45  Parallel  lines 59 

THE  CIRCLE,  A  SPECIAL  CASE  OF  THE  EQUATION  OF  THE   SECOND  DEGREE. 

46  The  general  equation  of  the  circle 61 

47  Intersections  of  circles 62 

48  Tangents  and  normals 63 

49  Condition   of  tangency 64 

50  Equation  of  the  tangent  through  a  given  point 65 

51  Sub-tangent  and  sub-normal 67 

52  Poles  and  polars  defined 68 

53  Equation  of  the  polar 69 

54  Co-ordinates  of  the  pole 70 

55  Polar  as  locus  of  harmonic  conjugates 70 

56  Polar  as  locus  of  poles 71 

ADDITIONAL  WORK   ON   THE   SUBJECT  OF  LOCI. 

57  General  remarks  on  loci  problems 74 

THE   GENERAL   EQUATION    OF    THE    SECOND   DEGREE. 

58  Nature  of  the  problem  and  of  the  method  employed 78 

59  The  "r"  equation 78 

60  One  chord  is  bisected  at  any  point 79 

61  Center  of  a  conic 79 

62  Diameters  of  a  conic 80 

63  Axes  of  symmetry  of  a  conic 80 

64  Reduction  of  the  general  equation 81 

THE    ELLIPSE    AND    THE    HYPERBOLA. 

65  Determination  of  form 84 

66  Early   geometric    definitions 86 

67  Mechanical    constructions 91 


Table  of  Contents. 

Article.  Page. 

68  Diameters 93 

69  Supplemental  chords -. 96 

70  Tangents  and  normals 97 

71  Poles  and  polars 99 

72  Asymptotes 99 

73  The  conjugate  hyperbola 101 

74  The  auxiliary  circles 102 

THE    PARABOLA. 

75  Determination  of  form 104 

76  Early   geometric   definition 104 

77  Mechanical   construction 105 

78  Diameters 106 

79  Tangents  and  normals 107 

80  Poles  and  polars 108 

ADDITIONAL   WORK   ON    THE   GENERAL   EQUATION    OF    THE    SECOND    DEGREE. 

81  Necessity  of  a  general  treatment , . .- 109 

82  Degenerate   conies 110 

83  Discriminant Ill 

84  Classification , '. .  112 

85  Tangents  and  normals 112 

86  A  second  condition  of  tangency 114 

87  Pole   and   polar ' 116 

88  Length   of  axes 117 

89  Foci   and   eccentricity 117 

90  Asymptotes 117 

91  Special  treatment  for  the  parabola 119 

92  Higher   Loci 122 

93  Families  of  conies 122 

OTHER    SYSTEMS    OF    CO-ORDINATES.       POLAR    CO-ORDINATES. 

94  Various  systems  of  co-ordinates 123 

95  Merits  and  demerits  of  various  systems 124 

96  Polar   co-ordinates 125 

97  The  relation  of  polar  and  cartesian  co-ordinates 127 

APPENDICES. 

A     Infinities  of  various  orders .* 129 

B     Functionality   131 

C     Permissible  operations 133 

D     Projection   135 

E     Imaginaries    138 


ANALYTIC  GEOMETRY. 

CHAPTER  I. 

Introductory. 

1.  The  student  who  begins  the  study  of  Analytic  Geometry  is 
assumed  to  have  acquired  previously  a  knowledge  of  elementary 
geometry  and  of  algebra. 

In  geometry  he  has  used  as  subject  matter  the  geometric  ele- 
ments (point,  line,  plane,  etc.)  and,  starting  from  certain  well 
defined  axioms  and  postulates,  has  deduced  the  properties  of  the 
simpler  plane  and  solid  geometric  forms.  He  has  also  developed 
a  method  of  investigation  which  enables  him  to  deal  with  questions 
of  form,  and  of  magnitude  so  far  as  it  depends  on  form.  In  algebra 
he  has  used  as  subject  matter  certain  symbols,  some  quantitative 
(x.  y,  a,  .  .  .  .  ),  some  operational  (+,  — ,  =,  V>  -^>  •  •)  ?  ^^^ 
has  developed  a  method,  far  more  general  than  that  of  arithmetic, 
of  dealing  with  questions  of  quantity. 

Each  of  these  methods  of  mathematical  investigation  (geometric 
and  algebraic)  has  its  advantages  and  its  limitations.  In  geom- 
etry we  work  with  elements  which  actually  possess  the  properties 
of  form,  position,  an(J  magnitude  in  which  we  are  interested,  and 
the  method  of  proof  is  in  consequence  frequently  suggested  by  a 
glance  at  the  diagram.  In  algebra  we  work  with  symbols  which 
have  in  themselves  no  properties  and  carry  no  power  of  suggestion. 
Consider  the  two  statements 

a.  Two  straight  lines  cannot  intersect  in  more  than  one  point ; 

b.  The  equations  2a?  —  3i/  +  1  =  0 
and                              4^  +  3t/  —  3  =  0 

cannot  be  simultaneously  satisfied  for  more  than  one  pair 

of  values  of  the  variables. 
These  statements  as  we  shall  see  later  are  practically  one  and 
the  same ;  but  the  first  is  evident  as  soon  as  the  diagram  is  drawn, 
while  the  second  requires  a  proof  of  whose  form  the  equations 
themselves  give  no  hint.  The  advantage  of  the  geometric  method 
is  evident. 


4  Analytic  Geometry. 

The  algebraic  method  on  the  other  hand  possesses  a  great  ad- 
vantage over  the  geometric  in  the  remarkable  generality  of  its 
processes  and  results.  It  includes  whole  classes  of  problems  in  a 
single  equation  and  expresses  the  solution  of  them  all  in  a  single 
formula.  For  example  all  quadratics  may  be  included  in  the 
single  form 

ax"  +  26iP  +  c  =  0, 

and  the  solutions  of  them  all  are  embraced  in  a  single  formula, 

V 


2.  Analytic  Geometry  is  an  attempt  to  establish  a  relation 
between  these  two  branches  of  mathematics,  so  that  the  methods 
of  either  may  be  applied  to  the  other;  in  short,  an  attempt  to 
establish  such  a  connection  that  one  may  write  the  formula  of  a 
curve  or  draw  the  diagram  of  an  equation. 

At  the  very  outset  we  are  confronted  by  a  difficulty.  Geometry 
deals  almost  wholly  with  fixed  objects,  definite  points,  lines,  and 
planes,  while  algebra  is  concerned  largely  with  variables.  But  we 
can  obviate  this  difficulty  by  thinking  from  now  on  of  all  curves 
as  traced  out  by  a  variable  point,  whose  variation  in  position  we 
shall  attempt  to  connect  with  the  change  in  value  of  algebraic 
variables. 


CHAPTER  II. 

ANALYTIC  GEOMETRY  OF  ONE  DIMENSION. 

3.  Consider  the  simplest  case   of  algebraic 

THE  SYSTEM  OF  variation.  A  single  variable  x  is  free  to  take 
CO-ORDINATES.  all  possible  values  from  minus  infinity  to  plus 

infinity.    An  equation  of  the  first  degree, 
X  —  a  :=  0, 

stops  the  variation  of  x  and  compels  it  to  take  the  value  a.  An 
equation  of  the  second  degree, 

{x  —  a)  (x  —  Z>)  =  x~ — {a-\-'b)x  -{-  ab=^^* 

compels  X  to  take  one  of  the  two  values  a  or  Z>;  and  similarly  for 
the  equations  of  the  higher  degrees.  Compare  all  this  with  the 
simple  case  of  geometric  variation  by  which  a  point  traces  out  a 
straight  line.  The  point  takes  an  infinite  number  of  dijfferent 
positions,  between  which  and  the  values  of  x  w^e  may  establish  a 
one  to  one  correspondence.  But  in  order  to  do  this  we  must  make 
certain  purely  arbitrary  assumptions. 

A.  We  must  assume  that  a  certain  point  on  the  line  is  to  repre- 
sent the  zero  value  of  x.  This  point  we  shall  call  the  origin.  Let 
it  be  the  point  0. 

0  ? 

Fig.  1. 

B.  In  order  to  represent  both  the  positive  and  negative  values 
of  X  we  must  divide  our  line  into  positive  and  negative  portions. 
We  assume  that  all  points  on  the  right  of  the  origin  correspond  to 
positive  values  of  x,  and  all  points  on  the  left  to  negative  values. 

C.  W>  must  decide  upon  a  scale  of  measurement ;  i.  e.,  whether 
we  will  measure  distances  on  the  line  in  feet,  inches,  millimeters, 
or  by  some  other  arbitrary  scale. 


*The  symbol  =  is  used  to  denote  an  identity.  The  student  must  be 
thoroughly  familiar  with  the  distinction  between  conditional  and  identical 
equations. 


6  Analytic  Geometry. 

Having  made  these  three  assumptions,  we  agree  that  the  point 
at  the  distance  a  (measured  by  the  agreed  upon  scale)  from  the 
origin  shall  correspond  to  the  value  a  of  x,  and  the  desired  corre- 
spondence between  the  point  and  the  variable  is  now  established. 
As  w  changes  continuously  from  minus  infinity  to  plus  infinity, 
the  point,  by  virtue  of  the  assumptions  we  have  made,  is  forced  to 
move  continuously  from  one  end  of  the  line  to  the  other,  while  to 
any  one  value  of  a)  there  corresponds  one  and  only  one  position  of 
the  point  and  conversely. 

The  distance  of  a  point  from  the  origin  is  called  the  co-ordinate 
of  the  point,  while  the  line  which  is  used  as  the  basis  of  the 
system  is  called  the  axis  of  co-ordinates,  or  merely  the  axis. 

4.  If  a  point  is  at  the  distance  a  from  the 

THE  RELATION  OF    origin  the  equation 

THE  POINT  AND  ^_^^or^_^_0, 

THE   EQUATION.  .  „     ,  ^.  ^.  ^  ^r,    ^        •    4.      mi  p 

is  called  the  equation  of  that  point.    Thus  if 

the  centimeter  is  the  unit  of  measure  the 

point  P  in  Fig.  1  is  spoken  of  as  the  point  4,  and  its  equation  is 

^  —  4  =  0. 
It  is  at  once  evident  that  every  first  degree  equation  in  one  variable 
represents  a  single  point,  and  conversely  that  any  point  on  the  line 
ma}'  be  represented  by  a  first  degree  equation  in  one  variable. 

The  second  degree  equation  in  one  variable  can  be  written  as 
the  product  of  two  factors,  and  is  satisfied  when  either  of  the 
two  factors  is  equal  to  zero.  It  is  accordingly  said  to  represent 
the  two  points  which  would  be  represented  by  the  two  factors 
taken  singly.    For  example, 

x^  —  3x-\-2  =  ix~2)ix  —  l)=0 
represents  the  two  points  at  distances  two  and  one  to  the  right 
of  the  origin. 

PROBLEMS. 

Locate  the  points  represented  by  the  following  equations : 


1. 

3a?  =  4 

2. 

.+  •5=5 

3. 

JL-1. 

4. 

*-«=^-10 

2        3 

2         2 

5. 

a?2  — 5a?  +  6  =  0 

6. 

2a?2_7^_2=:0 

7. 

x{x^-}-Ax  —  12)=0 

8. 

i2x^—10x  —  12){x'-- 

9. 

£i?2  — l  =  7a?  — 3 

10. 

a?3_4a?  =  0 

1)  =  0 


Analytic  Geometry.  7 

11.  Write  the  equation  which  represents  two  points  on  opposite 
sides  of  the  origin  and  at  a  distance  five  from  it. 

12.  If  the  unit  of  length  adopted  be  the  foot,  write  the  equations 
of  the  points  whose  distances  from  the  origin  are  respectively  3 
inches,  3  feet,  3  yards,  minus  1  yard. 

13.  Write  the  equation  representing  the  three  points  whose  co- 

1     4 
ordinates  are  a,  —  ,  — . 

'23 

14.  Generalize  the  work  in  problem  11 ;  i.  e.,  write  the  equation 
which,  if  proper  values  are  given  to  the  constants  involved,  will 
represent  any  pair  of  points  symmetrically  situated  with  respect  to 
the  origin. 

15.  Write  the  general  equation  representing  a  triad  of  points 
whose  co-ordinates  are  in  the  ratio  of  2,  3,  4. 

5.  The  equation 


THE   EQUATION 
WITH   EQUAL 
ROOTS. 


x^  —  2a30  -|-  a^  __  Q 

demands  special  consideration.     It  reduces 
at  once  to 

(x —  a)  {x  —  a)  ^  (a? —  a)^  =  0, 


and  we  might  say  that  the  equation  represents  only  the  point  at 
the  distance  a  from  the  origin.  But  this  mode  of  interpretation 
is  unwise  because  it  fails  to  recognize  any  distinction  between  the 
two  equations 

x^  —  2ax  +  a^  =  0 
and  X  —  a  =  0. 

We  shall  obtain  a  hint  of  the  proper  interpretation  if  we  ask 
how  the  equation  under  consideration  arises.  It  is  evidently  the 
limiting  form  of 

{x  —  a)  {x  —  6)^d?2 —  {a -{- h) X  -\-  aJ)  =  0 

as  h  tends  to  a  as  a  limit.    We  may  therefore  say, 

''x^  —  2ax  +  a-  =  0 

is  the  limiting  form  of  an  equation  representing  two  distinct 
points  as  the  points  tend  to  coincidence."  Mathematicians,  how- 
ever, are  accustomed  to  use  the  shorter  expression, 

''x^  —  2ax-i-a-  =  0 

represents  two  coincident  points."  understanding  by  this  exactly 
what  is  expressed  in  the  longer  phrase  above. 


8  '  Analytic  Geometry. 

The  method  of  interpretation  adopted  here  for  this  exceptional 
ease  of  the  second  degree  equation  is  a  general  method,  widely 
used  in  all  branches  of  mathematics,  and  the  student  should 
obtain  a  clear  understanding  of  it.    It  may  be  stated  as  follows : 

1.  Exceptional  cases  will  be  treated  by  regarding  them  as  limit- 
ing forms  of  the  more  general  case.  2.  If  any  expression  has  been 
used  to  represent  the  more  general  case,  the  limiting  form  of  that 
expression  will  be  used,  whenever  it  is  possible,  to  represent  the 
special  case.* 

6.  Another  type  of  equation  presents  a  more 

THE  EQUATION  serious  difficulty.    When  Z>-  —  ac  is  negative 

WITH  COMPLEX         the  equation 
ROOTS.t  ax^  +2h3o  +  c==0 

is  not  satisfied  by  any  real  value  of  x.    Con- 
sider the  special  case  of 

a?2  — 2a?  +  2  =  0. 
Here  a?  =  1  zt  V  —  1 

and  there  are  no  points  on  the  line  corresponding  to  these  values 
of  X.  The  difficulty  grows  out  of  the  nature  of  our  fundamental 
assumptions.  We  have  established  such  a  connection  between  the 
points  on  the  line  and  the  values  of  x  that  every  point  corresponds 
to  a  real  value  of  x  and  there  are  no  points  left  to  correspond  to 
imaginary  or  complex  values.  We  shall  accordingly  say  that  the 
equation  represents  a  pair  of  imaginary  points,  meaning  thereby 
merely  that  it  represents  a  pair  of  points  that  from  the  nature 
of  our  fundamental  assumptions  cannot  be  represented  in  our 
diagram. 

*As  another  illustration  consider  the  treatment  of  the  special  case  of 
division  by  zero.  It  is  regarded  as  the  limiting  form  of  division  by  a 
quantity  h,  as  6  tends  to  zero.  It  is  evident,  provided  the  dividend  is  not 
zero,  that  as  the  divisor  tends  to  zero    the    quotient    increases    indefinitely. 

Our  general   expression   for   division   is  —  =  c,    and  in  accordance  with  the 

second  part  of   our   principle  we  write  —  =  oo .    This   expression   must  not, 

however,  be  regarded  as  containing  any  statement  as  to  the  possibility  of 
dividing  something  by  nothing.  It  is  merely  a  short  hand  way  of  saying, 
"When  the  dividend  is  not  zero  and  the  divisor  tends  to  zero  as  a  limit 
the  quotient  increases  indefinitely."  Other  examples  of  this  mode  of  inter- 
pretation  will   be   met   with   from   time  to  time  in  our  work. 

tA  complex  number  is  one  of  the  form  a  -]-  b  y/  —  1 .  When  a  is  zero 
the  number  is  a  pure  imaginary,  when  6  is  zero  it  is  real.  From  now  on 
we  shall  denote  V  —  1  by  i  and  the  general  form  of  a  complex  number  will 
be  a  4-  ih. 


Analytic  Geometry.  9 

PROBLEMS. 
What  points  do  the  following  equations  represent? 

1.  x^  —  Sa}  —  28=0  .  2.  Sa?^  — 2j?  +  1  =  0 

3.   {x^  —  4.x  +  9){x  —  2)  =  0         4.   {x''  —  l){x^  +  l)=0 

5.  a?2(^2_3^_A  )=o  6.  x(x^—4.x-\-l)(x^+2)  =  0, 

4 

7.  In  building  our  system  of  co-ordinates  we 

TRANSFORMATION  made  three  purely  arbitrary  assumptions: 
OF  CO-ORDINATES  first,  that  the  origin  was  at  a  particular 
IN  ONE  DIMENSION,  point;  second,  that  distances  to  the  right 
were  to  be  counted  positive;  third,  that  dis- 
tances were  to  be  measured  by  a  given  scale.  A  change  in  any 
one  of  these  assumptions  will  of  course  lead  to  a  new  system  of 
co-ordinates,  and  the  algebraic  relation  between  the  two  systems 
must  be  determined. 

F         -R q 

Fig.  2. 

Suppose  for  example  that  the  point  P  has  been  taken  as  the 
origin,  distances  to  the  right  as  positive,  and  —  of  an  inch  as  the 

unit  of  measurement.  The  point  Q  will  then  have  the  co-ordinate 
10  and  be  represented  by  the  equation 

a?  — 10  =  0. 

If  now  we  take  as  a  new  origin  a  point  i2  at  a  distance  4  to  the 
right  of  P^  and,  to  avoid  confusion,  call  the  variable  in  this  new 
system  x',  we  see  at  once  that  the  old  and  new  co-ordinates  of  any 
point  are  connected  by  the  relation 

x=x'  -{-4t, 

and  that  the  new  equation  of  Q  is  therefore 

a?'  — 6  =  0. 

In  general  a  movement  of  the  origin  a  distance  h,  positive  or 
negative,  corresponds  to  the  algebraic  substitution 

x  =  x'  -}-  7i. 

If  we  change  our  assumption  as  to  the  directions  and  make 
distances  to  the  left  positive,  it  is  evident  that  the  corresponding 
substitution  is 

0}= — x\ 


10  Analytic  Geometry. 

If  we  change  our  assumption  as  to  the  unit  of  length,  and  re- 
place it  by  one  k  times  as  great,  it  is  evident  that  the  correspond- 
ing algebraic  substitution  is 

X  =  lex. 

If  we  note  that  the  substitution 

x  =  x  -\-}i 
followed  by  x= — x" 

followed  by  x"  =  kx'" 

gives  us  x  =  —  kx"'  +  h. 

we  see  that  the  substitution 

x  =  ax'  -{-h 

may  be  made  by  proper  choice  of  the  constants  a  and  1)  to  represent 
any  change  whatever  of  the  system  of  co-ordinates.  Any  such 
change  is  called  a  transformation  of  co-ordinates  and  the  corre- 
sponding algebraic  substitution  is  called  a  formula  of  trans- 
formation. 

The  mathematical  importance  of  the  subject  of  transformation 
of  co-ordinates  grows  out  of  the  power  which  the  corresponding 
substitutions  give  us  to  modify  equations.  For  example,  the  sub- 
stitution 

x  =  x'  -\-2 
reduces  x-  —  4:X  -\-S^^0 

to  the  simpler  form 

x'^  —  l  =  0, 
while  the  substitution 

x  =  x'  -{- 1 

reduces  the  same  equation  to 

x'^—2x=0. 
Again  the  equation 

9x^  —  Qx  —  S  =  0 

which  has  the  fractional  roots    -    and   —  -   is  reduced  by  the 

3  3 


substitution  x  =  -^ 

3 

to  the  form  x^^2x — 8  =  0 

which  has  the  integer  roots  4  and  2. 


Analytic  Geometry.  U 

PROBLEMS. 

Explain  the  geometric  significance  of  the  following  transfor- 
mations. 

1.  a?^a?'  +  2  2.  op  =  x' — 4 

3.  2x  =  2x  -\-S  4..  x  =  —2x' 

5.  x  =  —  3x'  —  4:  6.  2x=5  —  6x 

Write  the  formulae  of  transformation  which  shall  represent  the 
following  changes  of  the  system  of  co-ordinates : 

7.  A  movement  of  the  origin  a  distance  4  to  the  left. 

8.  A  movement  of  the  origin  to  the  point  whose  co-ordinate  is 
_  8^ 

2  ' 

9.  A  division  of  the  unit  of  length  by  4. 

10.  A  movement  of  the  origin  a  distance  2  to  the  right,  followed 
by  an  interchange  of  the  positive  and  negative  portions  of  the 
axis. 

11.  A  movement  of  the  origin  to  the  point  whose  co-ordinate  is 
4,  followed  by  a  multiplication  of  the  unit  of  length  by  three,  fol- 
lowed by  an  interchange  of  the  positive  and  negative  portions  of 
the  axis. 

8.  We  are  not  always  given  the  formula  of 

THE  DERIVATION  OF  transformation.     Cases  frequently  arise  in 

THE  FORMULA  OF  which  we  are  required  to  determine  the  sub- 

TRANSFORMATION  stitutiou  which  will  produce  a  given  form 

WHICH  WILL  when  applied  to  a  particular  equation.    For 

PRODUCE  A  GIVEN  example,  let  it  be  required  to  find  the  trans- 

RESULT.  formation  that  will  reduce  the  equation 

to  a  n«w  equation  which  has  no  term  of  the  first  degree  in  the 
variable.  Two  methods  of  solution  present  themselves.  We  may 
find  the  two  points  represented  by  the  equation  and  then  ac- 
complish the  desired  result  by  taking  the  point  midwaj^  between 
them  as  the  new  origin,  (Prob.  14,  Art.  4)  ;  or  we  may  assume  a 
general  substitution 

x^x  -\-  h, 

and  note  that  the  resulting  equation 

.2?'2  +  x'{2h  —  5)  +  7i2  —  57i  +  6  =  0 
has  no  term  of  the  first  degree  in  the  variable  if 

2h—5  =  0,  i.  e.,  ifh=  -^ 

2 
Our  substitution  is  therefore  determined. 


12  Analytic  Geometry. 

The  second  of  the  methods  used  is  the  more  general  and  there- 
fore of  course  the  more  valuable. 

PROBLEMS. 

Use  each  of  the  methods  outlined  above  to  determine  the  sub- 
stitutions which  will  reduce  the  following  equations  to  equations 
having  no  term  of  the  first  degree  in  the  variable : 
1.  a?2  +  3a?  —  7  =  0  2.  w"" —4.x —  1  =  0 

3.  a?2  — 5a?-f2==0  4.  a?=^  —  2a?  +  1  =  0. 

5.  Discuss  in  full  the  substitution  which  will  reduce  an  equation 
of  the  second  degree  to  an  equation  which  has  no  constant  term. 

6.  Show  that  no  movement  of  the  origin  can  change  the  degree  of 
the  equation. 


AND   ANHARMONIC 
RATIOS. 

in  the  ratio 


9.  If  we  are  given  two  points  A  and  B,  whose 

?[fT^.^.^.f .  ^f.T.'.?.^     co-ordinates  are  a?i  and  x^,  and  a  third  point 

C\  whose  co-ordinate  is  x^,  on  the  same  line, 

the  point  C  is  said  to  divide  the  segment  AB 

{x^—xy 

which  is  called  the  distance  ratio  of  C  with  respect  to  A  and  B. 

This  distance  ratio  is  evidently  numerically  equal  to  the  ratio  of 

4  C 
the  segments    - — -  and  is  positive  or  negative  according  as  the 

jBC 
point  C  is  within  or  without  the  segment  AB.    In  this  latter  fact 
lies  its  superiority  to  the  mere  numerical  ratio,  which  makes  no 
distinction  between  internal  and  external  division. 

If  a  fourth  point  D  whose  co-ordinate  is  x^  is  given  ,  the  ratio 
of  the  two  distance  ratios  so  determined 

X^         d?3 


a?,  — X. 


4  2 

i.  e., 

(d?i  — a?3)(j?,  — a?J 
(a?i  — a?J(a?3  — a?2) 

is  called  the  cross  ratio  or  the  anharmonic  ratio  of  the  four  points, 
and  is  denoted  by 

I        Xj^     a?2     a?3     a?4     I 


Analytic  Geometry.  13 

If  C  and  D  divide  the  segnient  AB  internally  and  externally  in  the 
same  ratio,  the  anharmonic  ratio  is  — '-  1,  the  division  is  said  to 
be  harmonic,  and  (J  and  D  are  said  to  be  harmonic  conjugates  with 
respect  to  A  and  B, 

PROBLEMS. 

1.  Find  the  distance  ratio  of  the  point  4  with  respect  to  the 
points  3  and  7. 

2.  Find  the  anharmonic  ratio  of  the  points  2  and  —  1  with 
respect  to  the  points  3  and  5. 

3.  Find  the  point  whose  distance  ratio  with  respect  to  the 
points  2  and  4  is  —  5. 

4.  Find  a  point  x  such  that  the  anharmonic  ratio  of  7  and  x  with 
respect  to  4  and  5  shall  be  2. 

5.  Find  the  harmonic  conjugate  of  the  origin  with  respect  to 
4and  — 1. 

6.  Generalize  problem  3,  i.  e.,  find  the  point  whose  distance  ratio 
w^ith  respect  to  x^  and  x.-,  shall  be  k. 

7.  Show  that  the  definition  given  above  of  harmonic  division  is 
equivalent  to  the  following:  The  segment  AB  is  divided  harmon- 
ically by  P  and  Q  if   —  +  — !—  =  ^-  . 

QA^  QA       QP 


CHAPTER  III. 

SOME  FUNDAMENTAL  IDEAS  OF  THE   ANALYTIC 
GEOMETRY  OF  TWO  DIMENSIONS. 

lO-  .     We  have  built  up  in  the  preceding  pages 

THE  SYSTEM  OF        an  analytic  geometry  of  one  dimension  in 

CO-ORDINATES.  order  to  illustrate  the  variation  of  a  single 

algebraic  variable.    Let  us  now  consider  the 

less  simple  case  of  two  variables. 

Two  variables,  x  and  y,  are  each  free  to  take  any  value  from 
minus  infinity  to  plus  infinity  and  these  values  are  paired  (one  of 
X  with  one  of  ^)  in  all  possible  ways.  We  desire  to  establish  a  one 
to  one  correspondence  between  these  pairs  of  values  and  the  mem- 
bers of  some  group  of  geometric  objects,  just  as  we  established  a 
one  to  one  correspondence  between  the  values  of  one  variable  and 
the  points  of  a  straight  line.  Since  with  any  one  of  the  infinite 
number  of  values  of  x  may  be  paired  any  one  of  the  infinite 
number  of  values  of  y,  the  total  number  of  pairs  is  infinity  squared, 
or  to  use  a  more  satisfactory  phrase,  is  doubly  infinite,  or  an 
Infinitj^  of  the  second  order.*  The  number  of  points  in  a  line 
however  is  singly  infinite,  and  our  former  mode  of  representation 
therefore  fails  us,  since  we  cannot  establish  a  one  to  one  corre- 
spondence between  a  singly  infinite  number  of  points  and  a  doubly 
infinite  number  of  pairs  of  values.  The  plane  however  contains 
a  doubly  infinite  number  of  points  and  may  therefore  be  used. 

In  order  to  establish  this  correspondence  between  the  pairs  of 
values  of  x  and  y  and  the  points  of  the  plane  we  must  make  certain 
arbitrary  assumptions. 


^See  Appendix  A.     Infinities  of  various  orders. 


Analytic  Geometry.  15 

A.  We  assume  two  intersecting  straight  lines  as  a  basis  of 
reference.     Let  them  be  OA  and  OB. 

B.  We  assume  positive  and  neg- 
ative directions  on  these  two  lines. 
Let  OA  and  OB  be  positive,  and  OA' 
and  OB'  negative. 

C.  We  assume  that  values  of  v 
correspond  to  distances  from  the  line 
BB'  measured  (by  any  desired  unit 

^  of  length)  parallel  to  the  line  AA'. 

D.  We  assume  that  values  of  y 
j,j^  2                       correspond  to  distances  from  the  line 

*  A  A'  measured  (by  any  desired  unit 
of  length)  parallel  to  the  line  BB'.  (The  units  determined  by  the 
assumptions  G  and  D  will  usually  but  not  always  be  the  same.) 

The  lines  AA'  and  BB'  are  called  Axes  of  Co-ordinates.  AA' 
is  the  X  axis  or  axis  of  abscissas.  BB'  is  the  Y  axis  or  axis  of 
ordinates.  The  intersection  of  the  axes  is  the  origin.  The  distance 
of  a  point  from  the  Y  axis,  measured  parallel  to  the  X  axis,  is 
called  the  abscissa,  or  the  x  co-ordinate,  or  frequently  merely  the  w 
of  the  point.  The  distance  of  a  point  from  the  X  axis,  measured 
parallel  to  the  Y  axis,  is  called  the  ordinate,  or  the  y  co-ordinate, 
or  frequently  merely  the  y  of  the  point.  Thus  in  the  diagram  the 
point  P  has  the  abscissa  a^  and  the  ordinate  Z>.  The  point  P  is 
frequently  spoken  of  as  the  point  {a^  h).  In  this  notation  the  x 
co-ordinate  is  always  the  one  first  mentioned.  When  the  axes  are 
perpendicular  to  each  other  the  co-ordinates  are  called  rectangular 
co-ordinates,  otherwise  they  are  oblique  co-ordinates.  From  now 
on  we  shall  understand  rectangular  co-ordinates  to  be  used  unless 
it  is  otherwise  stated.* 

The  four  assumptions  made  above  establish  a  complete  corre- 
spondence between  the  points  in  the  plane  and  all  pairs  of  real 
values  of  x  and  y.  To  each  point  corresponds  one  and  only  one 
pair  of  values  and  conversely. 


*Siich  a  system  of  co-ordinates  as  Ave  have  outlined  above  is  frequently 
called  Cartesian,  in  honor  of  Descartes,  who  was  the  first  worker  in  this 
field.     Consult  Ball's   Short  History  of  Mathematics   or   some   similar  work. 


16  Analytic  Geometry, 

PROBLEMS. 

1.  Locate  the  following  points:  (1,  1),  (—1,  —1),  (0,  3), 
(2,  0),  (-5,  3),  (0,  0),  (5,  -l),(-2,  4^^,  g,  -J),  (-2,  -4), 

(0,-5).  -^       ^ 

2.  Tfce  lower  left  hand  vertex  of  a  square  whose  side  is  4  is  at 
the  origin  and  two  sides  coincide  with  the  axes.  Find  the  co- 
ordinates of  the  other  three  vertices. 

3.  The  upper  right  hand  vertex  of  a  rectangle  whose  sides  are 
8  and  3  is  at  the  origin  and  its  longer  side  coincides  with  the  X 
axis.    Find  the  co-ordinates  of  the  middle  points  of  the  sides. 

4.  The  base  of  an  equilateral  triangle  whose  side  is  4  is  parallel 
to  the  X  axis,  and  the  point  (0,  2)  is  the  middle  point  of  the  base. 
Find  the  co-ordinates  of  the  three  vertices. 

5.  The  axes  are  the  diagonals  of  a  square  whose  side  is  8.  Find 
the  co-ordinates  of  the  vertices. 

6.  A  regular  hexagon  whose  side  is  5  has  its  center  at  the 
origin  and  one  pair  of  vertices  on  the  X  axis.  Find  the  co- 
ordinates of  the  remaining  vertices. 

Il-  If  a?  and  y  are  subject  to  no  restriction  the 

THE  SIGNIFICANCE  corresponding  point  may  take  anv   position 

OF  TWO  in  the  plane.    If,  however,  we  are  given  two 

SIMULTANEOUS  equations  which  the  variables  must  satisfy. 

EQUATIONS.  the  point  is  no  longer  free.    For  example 

Sx  —  y  —  5=0 

and  ^x-\-y  — 11  =  0 

are  both  satisfied  only  when  we  have 

x^2,y=l. 

These  two  equations  therefore  restrict  the  varying  point  to  the 
single  position  (2,  1),  and  may  therefore  be  said  to  represent  this 
point.    Again 

^'  +  2/'  — 6^  +  4=0 
and  X  -\-y=^{) 

may  be  said  to  represent  the  two  points  (1, 1)  and  (2,  2)  since  only 
for  these  values  of  x  and  y  can  these  two  equations  be  simulta- 
neously satisfied. 

Since  two  algebraic  equations  in  two  variables  always  on  solu- 
tion give  a  finite  number  of  pairs  of  values  of  the  variables,  it 
follows  that  two  algebraic  equations  in  two  variables  restrict  the 
varying  point  to  a  finite  number  of  fixed  positions. 


Analytic  Geometry.  17 

PROBLEMS. 

Find  the  points  rej)resented  by  the  following  pairs  of  simulta- 
neous equations : 

1.  4.x— y +  22  =  0  2.  3a?+i/  +  2  =  0 
x  —  2y-^d  =  0  Qx  —  3y—ll  =  0 

3.  4^  — I/  — 4  =  0  4.  a)-\-2y  =  () 
5x-^2y  —  5  =  0  3^— 4i/  =  0 

5.  x^-\-y'  —  9  =  0  6.  y^  —  8x  =  0 
iP  +  27/  — 1  =  0  x  +  2y  =  0 

12.  When  we  have  a  single  equation  the  values 

THE  SIGNIFICANCE  of^  and  y  are  no  longer  determined,  but  are 
OF  A  SINGLE  nevertheless  not  entirely  free.    The  equation 

EQUATION.    LOCI.       acts  as  a  restriction  on  the  movement  of  the 
variable  point,  without  fixing  its  position. 
For  example 

w  —  y^O 

is  satisfied  by  the  points*  (1, 1) ,  (2,  2) ,  (3,  3)  but  not  by  the  points 
(1,  2),  (1,  4),  (2,  3).  In  fact  it  is  satisfied  by  every  point  whose 
co-ordinates  are  equal  and  of  the  same  sign  and  by  no  others. 
It  is  therefore  satisfied  by  all  points  on  that  bisector  of  the 
angle  between  the  axis  which  passes  through  the  first  and  third 
quadrants,  and  by  no  others.  This  bisector  is  therefore  called  the 
locus  of  points  subject  to  the  given  condition,  and 

X  —  y^O 

is  called  the  equation  of  the  bisector,  or  the  equation  of  the 
locus.  In  the  same  way  any  equation  between  x  and  y  acts  as  a 
restriction  on  the  variation  of  the  point  which  has  x  and  y  as  its 
co-ordinates.  The  aggregate  of  all  points  which  satisfy  a  given 
condition  is  said  to  constitute  a  locus,  and  the  equation  which 
expresses  the  condition  is  called  the  equation  of  the  locus.  The 
student  may  accept  without  proof  for  the  present  the  statement 
that  the  locus,  as  used  in.  analytic  geometry  of  two  dimensions, 
consists  in  general  of  one  or  more  lines,  straight  or  curved. 

It  sometimes  happens  that  the  left  hand  member  of  the  given 
equation  is  the  product  of  two  or  more  rational  factors.  In  this 
case  it  is  evident  that  the  equation  will  be  satisfied  by  such 


"This    is    the   usual    abbreviated   form    for   the    expression,    "The    equation 
X  —  y  =  0   is   satisfied  by  the  co-ordinates  of  the  points,  '  etc. 


18  Analytic  Geometry. 

points  as  make  any  one  of  the  factors  equal  to  zero  ;  i.  e.,  the 
locus  in  this  case  consists  of  two  or  more  parts,  each  of  which 
would  be  represented  by  equating  one  of  the  factors  to  zero. 
Such  a  locus  is  said  to  be  a  degenerate  locus.     For  example 

is  satisfied  by  all  points  on  the  bisector  through  the  first  and  third 
quadrants,  but  also  by  the  points  on  the  bisector  through  the 
second  and  fourth  quadrants.  The  equation  is  said  therefore  to 
represent  the  degenerate  locus  consisting  of  these  two  bisectors. 

This  relation  between  equation  and  locus  is  the  fundamental 
idea  in  our  present  subject.  The  problems  to  which  it  gives  rise 
may  usually  be  divided  into  the  following  groups : 

a.  To  construct  the  locus  when  the  equation  is  given. 

b.  To  construct  the  locus  when  a  finite  number  of  points  upon 
it  is  given. 

c.  To  deduce  the  equation  when  the  restrictions  on  the  move- 
ment of  the  tracing  point  are  given. 

d.  To  deduce  the  equation  when  a  finite  number  of  points  on  a 
locus  of  some  known  type  is  given. 

e.  To  deduce  the  geometric  properties  and  relations  of  loci  from 
a  consideration  of  the  corresponding  equations. 

These  problems  we  shall  now  proceed  to  treat  in  turn. 


CHAPTER  IV. 

TO  CONSTRUCT  THE  LOCUS  WHEN  THE  EQUATION  IS 

GIVEN. 

13.  It  sometimes  happens,  as  in  the  case  dis- 

THE  PROCESS  OF  cussed  at  the  opening  of  the  previous  para- 
CURVE  PLOTTING,  graph,  that  the  equation  is  so  simple  that 
the  form  of  the  h)cus  can  be  at  once  inferred 
from  the  equation  ;  and  the  ability  to  recognize  in  this  way 
a  large  number  of  simple  loci  can  easily  be  acquired. 

In  more  complex  cases  it  is  not  so  easy  to  infer  the  form  of  the 
locus  from  the  equation,  and  the  following  method  is  adopted. 
Assume  in  succession  a  number  of  values  of  one  variable,  sub- 
stitute these  in  the  equation  and  compute  the  corresponding 
values  of  the  other.  Each  of  the  pairs  of  values  thus  determined 
will  locate  a  point  on  the  locus.  By  locating  a  suflScient  number 
of  such  points  the  form  of  the  locus  may  be  inferred.  For  ex- 
ample, consider  the  equation 

x—Sij  +  2  =  0. 

If  we  substitute  in  succession  for  x  in  this  equation  the  integer 
values  from  —  7  to  +  7,  and  compute  the  corresponding  values 
of  y^  we  obtain  the  following  points  which  satisfy  the  equation : 

(-^.  -|),  (-6,  -|  ),  (-5,  -1),  (-4,  -|),  (-3,-i  ), 
(-2,0),  (-1,  i),  (0,  |),  (1,  1),  (2,  |),  (3,  |),  (4,  2), 
(5,   J),  (6,  |),  (7,3). 

If  we  locate  these  points  on  our  diagram  we  shall  find  that  they 
all  lie  on  a  straight  line,  and  consequently  we  may  infer  that  this 
line  is  the  locus.  But  in  drawing  this  inference  we  make  two 
assumptions:  first,  that  the  point  which  traces  the  locus  moves 
continuously  and  not  by  leaps  from  point  to  point,  or  in  other 


Analytic  Geometry. 


21 


i5f 


words  that  the  locus  is  a  continuous  curve  and  not  a  number  of 

separate  fragments  or  isolated  points;  second,  that  the  tracing 

point  moves  along  the  curve  AB 

rather  than  along  some  other 

curve  through  the  same  points, 

as  the  undulating  curve  in  the 

figure. 

We  can  practically  convince 
ourselves  of  our  right  to  make 
these  assumptions  in  this  par- 
ticular case  by  taking  values  of 
w  between  those  already  taken, 
and  in  this  way  finding  addi- 
tional points  of  the  locus. 
The  mathematical  treatment  of 
these  difficulties  must  be  de- 
ferred to  a  later  point  in  the  student's  career. 

As  a  second  example,  consider  the  equation 
Solve  the  equation  for  y  and  we  have 


Fig.  4. 


U 


V9 


We  see  at  once  that  it  is  useless  to  look  for  points  whose  w  does 
not  lie  between  —  3  and  +  3,  since  any  value  of  x  outside  these 
limits  gives  y  imaginary  values,  and  our  fundamental  assumptions 
are  such  that  a  pair  of  values,  one  or  both  of  which  are  imaginary, 
has  no  corresponding  point  in  the  plane.  Giving  w  a  series  of 
values  between  —  3  and  +  3,  and  computing  the  corresponding 
values  of  y,  we  obtain  the  following  points  which  satisfy  the 
equation 

(-3,0),(-2,±|-v-5),  (-1,  ±^i/2),  (0,  ±2),    (1,  ±|v^2), 


(2,  ±|-t/5),  (8,  0). 


22 


Analytic  Geometry. 


Plotting  these  points  we  readily 
infer  the  curve  to  be  of  the  form  in 
the  figure,  an  inference  which  we 
may  confirm  as  in  the  last  example. 
It  is  sometimes  more  convenient 
to  assume  arbitrary  values  for  y  and 
compute  the  corresponding  values 
of  X,  as  for  example  in  the  equation 

x=if  —  Zy  +  2. 
The  variable  to  which  arbitrary  val- 
ues are  assigned  is  called  the  independent  variable,  the  other 
the  dependent  variable.  The  distinction  is  evidently  a  purely 
arbitrary  one  and  the  student  will  in  each  case  so  choose  the 
independent  variable  as  to  make  his  work  as  simple  as  possible, 
14. 


Fig,  5. 


UTILITY  OF  THE 
PROCESS. 


The  process  illustrated  in  the  last  para- 
graph, and  usually  spoken  of  as  curve  plot- 
ting, frequently  enables  us  to  visualize  a 
formula  and  obtain  a  clearer  idea  of  its  sig- 
nificance than  we  could  obtain  in  any  other  manner.  For 
example,  consider  the  case  of  a  body  of  mass  m  moving  with  a 
velocit;^  v.  Its  momentum  M  and  its  kinetic  energy  E  are  given  by 
the  formulae  M  =  mv 


and 


E  =  —  mv^. 


A  study  of  these  formulae  will  give  the  student  some  idea  of 
the  relative  variation  of  momentum  and  energy  as  the  velocity 
changes,  but  a  much  clearer  idea  will  be  obtained  by  an  examin- 
ation of  Fig.  6,  where  the  corresponding  curves  are  plotted  on  the 
same  axes.  In  each  case  the  horizontal  axis  is  chosen  as  the 
axis  of  velocities,  and  in  order  to  make  the  problem  a  definite 
one  the  mass  m  is  taken  as  unity.      ' 

PROBLEMS. 

Plot    the  following  curves,  inferring,  whenever  you  can,  the 

form  of  the  curve  directly  from  the  equation.* 

1.  a?  =  0  2.  y  =  {) 

3.  2/  =  2  4.  3a?  =  6 

5.  £P  +  i/  =  0  6.  a?  — 2^  =  0 

7.  2.a?  +  4t/  =  0  8.  x''+y^  =  4t 

9.  a?2_j_27/  =  0  10.  a?2+i/2_2£p  =  0 

11.  a?  +  t/  — 1  =  0  12.  y  =  x''  -\-Zx~l 

*The  student   should   use   co-ordinate  paper   ruled  to  tenths. 


Analytic  Geometry.  23 

13.  The  distance  passed  over  by  a  moving  body  in  t  seconds 
is  given  by  the  formula 

€1"=  Vot  -{-  —  af, 

where  ij„  is  the  initial  velocity  and  a  is  the  acceleration.    Plot  the 
curves  for  the  three  special  cases 

(1)  ^0=0,  a  =  2, 

(2)  Vo  =  2,  a  =  0, 

(3)  Vo  =  2,  a  =  2, 
and  compare  them. 

14.  The  intensity  of  the  light  at  any  point  varies  inversely  as  the 
square  of  the  distance  of  that  point  from  the  source  of  light ;  i.  e., 

i  =  —,.    Let  k  equal  one,  and  plot  the  corresponding  curve. 

15.  The  intensity  of  the  magnetic  field  in  the  vicinity  of  a  wire 
carrying  an  electric  current  varies  inversely  as  the  distance  from 

the  wire;  i.  e.,  i=  — .    Let  k=l,  plot  the  corresponding  curve, 

and  compare  it  with  the  results  of  the  last  problem. 

16.  Show  that  the  locus  represented  by  the  equation 

aj?  +  Z>i/  +  c  =  0    . 

can  have  no  point  in  the  first  quadrant  if  a,  1),  and  c  are  all 
positive. 

17.  State  a  corresponding  theorem  for  the  third  quadrant. 


CHAPTER  V. 

TO  CONSTRUCT  THE  LOCUS  WHEN  A  FINITE  NUMBER 
OF  POINTS  UPON  IT  IS  GIVEN. 


15. 


THE   METHOD  OF 
PLOTTING. 


The  equation  of  the  locus  is  not  always 
given.  The  law  w^hich  regulates  the  phe- 
nomena may  not  be  known  or  may  be  too 
complex  for  simple  mathematical  expression. 
We  may  however  still  have  sufficient  data  to  plot  the 
curve.  For  example,  a  student  in  the  laboratory  desires 
to  study  the  rate  at  which  a  body  cools.  He  may  not  know  the 
formula  Avhich  connects  the  temperature  of  the  body  with  the 
length  of  time  it  has  been  allowed  to  cool ;  but  he  records  the 
temperature  at  short  in.tervals  and  secures  in  this  way  a  number  of 
pairs  of  values  of  time  and  temperature  which  he  may  regard  as 
co-ordinates,  and  thus  plot  a  number  of  points  on  the  curve.  This 
curve  (Fig.  7,  p.  20)  shows  clearly  the  relation  between  temper- 
ature and  time  of  cooling. 

Again,  the  number  of  students  in  a  given  institution  from  year 
to  year  depends  on  too  many  causes  for  the  relation  to  be  ex- 
pressed by  an  equation.  But  the  curve  can  be  plotted  and  thus  a 
clear  and  condensed  representation  of  the  variation  can  be  secured. 
The  attendance  at  Johns  Hopkins  University  from  1877  to  190i  is 
given  in  the  subjoined  table. 


1877 

89 

1884 

1878 

104 

1885 

1879 

123 

1886 

1880 

159 

1887 

1881 

176 

1888 

1882 

175 

1889 

1883 

204 

1890 

249 
290 
314 
378 
420 
394 
404 


1891 
1892 
1893 
1894 
1895 
1896 
1897 


468 
547 
551 
522 

589 
596 
520 


1898 

641 

1899 

649 

1900 

645 

1901 

651 

1902 

694 

1903 

695 

1904 

715 

A  glance  at  the  corresponding  curve  (Fig.  8)  giv^es  however  a 
much  clearer  picture  of  the  growth  of  the  institution. 


Analytic  Geometry. 


25 


PROBLEMS. 

1.    The  folloAving  table  gives  the  number  of  years  one  may  expect 


to  live  at  the  ages  indicated : 


Age 

Expectation 

Age 

Expectatio7i 

^9 

0 

39.9 

35 

29.4 

70 

5 

49.1 

40 

26.0 

75 

10 

47.0 

45 

22.7 

80 

15 

43.1 

50 

19.5 

85 

20 

39.4 

55 

16.4 

90 

25 

30.1 

60 

13.5 

95 

30 

32.7 

65 

10.8 

100 

Age  Expectation 
8.4 
6.4 
4.9 
3.7 
2.8 
2.1 
1.6 


riot  the  corresponding  curve. 

2.  The  temperature  of  a  fever  patient  was  as  follows: 


July    8, 

5 

P.  M. 

99.4 

Julv 

13, 

12.30 

P.  M. 

106.3 

July    9, 

6 

A.M. 

98.0 

July 

13, 

1.30 

P.  M. 

105.6 

July    9, 

5 

P.M. 

105.0 

July 

13, 

5 

P.  M. 

104.6 

July  10, 

6 

A.  M. 

99.0 

July 

14, 

6 

A.M. 

98.2 

July  10, 

5 

P.  M. 

99.2 

July 

14, 

5 

P.M. 

98.6 

July  11, 

6 

A.M. 

98.2 

July 

15, 

6 

A.M. 

98.0 

July  11, 

1 

P.  M. 

106.0 

July 

15, 

5 

P.  M. 

99.0 

July  11, 

2 

P.  M. 

104.0 

July 

16, 

6 

A.M. 

98.0 

July  11, 

5 

P.M. 

103.6 

July 

16, 

5 

P.  M. 

98.4 

July  12, 

6 

A.M. 

98.0 

July 

17, 

6 

A.  M. 

98.0 

July  12, 

5 

P.  M. 

98.4 

July 

17, 

5 

P.M. 

98.6 

July  13, 

6 

A.  M. 

99.0 

July 

18, 

6 

A.M. 

98.0 

Plot  the  corresponding  curve.  (In  this  problem  it  will  evi- 
dently be  wise  to  take  as  the  axis  of  X  the  line  corresponding  to 
some  high  temperature,  such  as  the  normal  temperature  of  98 
degrees,  in  place  of  the  line  corresponding  to  zero  degrees.) 

4.  Select  some  prominent  stock  or  article  of  produce  and  plot 
its  prices  for  the  next  two  weeks  as  given  in  the  daily  stock  jr 
market  reports,  giving  the  reasons  for  any  important  fluctuations. 

16.  In  these  examples  there  is  no  particular 

CHOICE  OF  SCALES  reason  why  the  same  scale  should  be  used 
OF  MEASUREMENT,  on  both  axes.  The  fact  that  a  certain  dis- 
tance has  been  used  to  denote  a  year  or  a 
day  is  no  reason  w^hy  the  same  distance  should  be  used 
to  denote  a  dollar  or  a  degree  of  temperature.  Mathe- 
maticians are  consequently  accustomed  to  select  such  scales 
as  are  most  convenient,  and  to  indicate  their  choice  by  a  foot 


26  Analytic  Geometry. 

note  as  in  Fig.  6,  or  by  figures  on  the  axes  as  in  Figs.  7  and  8, 
The  choice  is  lisually  so  made  as  to  make  the  important  features 
of  the  curve  prominent.  Thus  in  problem  2  the  physician  will 
choose  a  small  distance  to  represent  the  day  and  a  much  greater 
one  to  represent  the  degree ;  while  the  curv  e  of  problem  1  may  be 
almost  wholly  deprived  of  interest  by  the  choice  of  a  large  dis- 
tance to  represent  the  unit  of  age  and  a  small  one  to  represent 
the  unit  of  expectation.  It  is  sometimes  necessary  after  a  curve 
has  been  drawn  to  one  scale  to  make  magnified  drawings  of  certain 
portions  in  order  to  examine  more  closely  certain  doubtful  points. 

17.  When  both  the  assumptions  made  in  para- 

DETERMINATION  graph  13  can  be  granted,  pairs  of  values 
OF  INTERMEDIATE  intermediate  between  those  actually  known 
VALUES.  can  be  determined  with  a  close  degree  of 

approximation  by  measuring  the  co-ordi- 
nates of  the  corresponding  point  on  the  curve.  Thus  in  problem  1 
of  paragraph  15  the  expectation  for  any  intermediate  age  can  be 
determined  by  measuring  the  ordinate  corresponding  to  the  ab- 
scissa representing  that  age.  But  in  many  cases  this  method  of 
procedure  is  not  in  order  since  the  assumptions  of  paragraph  13 
cannot  be  made.  In  the  Johns  Hopkins  problem  for  example  the 
attendance  does  not  pass  continuously  from  520  to  641,  taking 
all  the  intermediate  values,  but  passes  by  leaps  from  one  value 
to  another.  In  other  words  the  actual  locus  is  not  a  continuous 
line  as  we  have  drawn  it,  but  a  succession  of  disconnected  points. 
The  second  problem  of  paragraph  15  is  a  case  in  which  the 
second  assumption  of  paragraph  13  cannot  be  granted.  While 
it  is  true  that  the  temperature  varies  continuously  with  the 
time,  it  is  by  no  means  to  be  expected  that  our  curve,  based  upon 
observations  made  at  intervals  of  several  hours,  shows  all  these 
variations.  There  may  be  and  probably  are  intermediate  vari- 
ations of  which  we  have  no  record.  In  such  a  case  as  this  where 
the  nature  of  the  phenomena  is  not  sufficiently  well  understood 
to  enable  us  to  deny  the  existence  of  these  intermediate  vari- 
ations, as  we  might  in  the  age-expectation  problem,  and  where 
the  laAvs  regulating  the  phenomena  do  not  admit  of  algebraic 
expression,  there  is  of  course  no  way  of  determining  the  number  or 
location  of  such  intermediate  variations.  When  the  locus  is 
determined  by  an  equation  they  may  be  determined  by  the  aid 
of  the  differential  calculus. 


Analytic  Geometry.  27 

18.  The  question   of  continuity,   raised   inci- 

DEFINITION  AND        dentally    in    the    last    paragraph,    demands 
TEST  OF  careful  consideration.    In  order  that  a  curve 

CONTINUITY.  may  be  continuous  it  is  necessary  that  as  x 

(the  abscissa  of  the  tracing  point)  varies 
continuously,  y  (the  ordinate  of  the  tracing  point)  shall  also  vary 
continuously ;  or,  expressed  algebraically,  1/  is  a  continuous  func- 
tion of  X  when  the  change  in  y  due  to  a  change  in  x  may  be  made 
as  small  as  we  please  by  taking  the  change  in  x  small  enough.  At 
all  points  where  this  condition  is  satisfied  y  is  said  to  be  a  con- 
tinuous function  of  x,  at  any  point  where  it  is  not  satisfied,  y  is 
said  to  be  discontinuous.  The  fever  temperature  problem  affords 
us  an  example  of  a  continuous  function.  If  the  change  in  the  time 
be  small  enough  the  change  in  the  patient's  temperature  will  be  as 
small  as  we  please,  while  in  the  Johns  Hopkins  problem  no 
shortening  of  the  interval  will  make  the  difference  between  two 
successive  values  of  y  a  small  quantity. 

Let  us  examine  the  continuity  of  a  simple  function,*  say 

y  =  2^.  t 

If  X  increases  by  an  amount  A/.%  or  as  we  more  frequently 
say,  takes  an  increment  Az?,  //  takes  an  increment  which  we 
may  call  Ay. 
Then  i/  =  2x' 

.    ?/  + A]/  =  2(a?  + ASy-=2a^'  +  4a?Z^i  +  2  AS' 

therefore  Ay  =  ^x /\x  -\- 2  Ax  - 

So  long  as  x  remains  finite,  the  right  hand  side  of  this 
equation  tends  to  zero  as  A^?  tends  to  zero,  or  in  other  words 
Ay  may,  by  a  proper  choice  of  A^,  be  made  as  small  as  we 
please  for  all  finite  values  of  x.  That  is,  //  is  a  continuous 
function  of  x  so  long  as  x  is  finite. 


*If  the  student  is  not  familiar  with  the  ideas  and  notation  of  mathe- 
matical  functionality,   he   should   at   this  time   read  Appendix  B. 

tThe  method  here  used  is  applicable  to  the  most  complex  forms,  but 
numerous  algebraic  difficulties  are  encountered  in  the  attempt  to  employ  it. 
The  overcoming  of  these  difficulties  falls  in  the  province  of  the  differential 
calculus. 


28  Analytic  Geometry. 

19-  The  definition  of  continuity  given  above 

INFINITE  brings  to  light  also  another  sort  of  discon- 

DISCONTINUITY.         tinuitv.     If  the  student  Avill  plot  the  curve 

he  will  note  that  as  x  tends  to  unity,  y  passes  beyond  all 
limit  (i.  e.,  becomes  infinite),  but  so  long  as  x  differs 
ever  so  little  from  unity  y  remains  finite.  That  is,  no  nmtter  how 
small  the  increment  which  carries  x  from  its*  previous  value  to 
the  value  unity,  y  leaps  from  a  finite  to  an  infinite  value.  The 
function  is  accordingh'  said  to  have  an  infinite  discontinuity, 
while  the  discontinuities  previously  discussed  are  called  finite 
discontinuities.  Such  functions  as  the  student  will  meet  in  the 
present  work  are,  as  may  be  shown  by  the  calculus,  free  from  finite 
discontinuities,  and  such  infinite  discontinuities  as  may  occur 
can  be  detected  by  plotting  the  curve.* 

PROBLEMS. 

Find  the  values  of  x  for  which  the  following  functions  have 

infinite  discontinuities. 

x~2 
1.  y  = 


J-  — 3 


2.  y  = 


{x~l)ix  —  2) 
S.  y^  sec  w 


*The  Johns  Hopkins  problem  and  problem  4  of  paragraph  15  furnish 
examples  of  finite  discontinuities,  but  in  all  these  cases  the  functional  re- 
lation is  not  given,  so  that  they  constitute  no  exception  to  the  statement 
made  above.  As  an  example  of  a  function  with  a  finite  discontinuity  con- 
sider the  equation 

As  X  (regarded  as  positive)  tends  to  zero,  y  tends  to  zero;  but  as  a?  (re- 
garded as  negative)  tends  to  zero,  y  tends  to  unity.  The  corresponding  locus 
has  therefore  a  finite  jump  from  unity  to  zero  as  it  crosses  the  Y  axis. 


CHAPTER  VI. 

TO  DEDUCE  THE  EQUATION  WHEN  THE  RESTRICTIONS 
ON  THE  MOVEMENT  OF  THE  TRACING  POINT  ARE 

GIVEN. 

20.  Problems  of  this  type  by  no  means  always 

THE  GENERAL  occur  in  the  simple  form  indicated  in  the 

METHOD.      .  heading   of  this   chapter.     Very  frequently 

we  are  confronted  with  a  mere  verbal  de- 
scription of  a  finished  curve,  containing  apparently  no  refer- 
ence to  the  method  of  its  construction.  But  if  the  ver- 
bal description  is  complete  it  contains  all  the  limitations 
on  the  movement  of  the  tracing  point.  The  method  of  meeting 
the  problem  is  therefore  always  the  same.  Consider  the  curve  as 
traced  by  a  variable  point;  determine  the  law  which  regulates 
the  movement  of  the  point,  (i.  e.,  the  condition  which  is  satisfied 
by  all  points  on  the  curve  and  by  no  others)  ;  state  this  condition 
in  algebraic  form,  i.  e.,  express  it  as  a  relation  between  the 
variable  co-ordinates,  x^  y,  of  the  variable  point.  We  have  then 
an  equation  of  condition  between  x  and  y  which  is  satisfied  by  the 
co-ordinates  of  all  points  on  the  curve  and  by  no  others;  in  other 
words  we  have  the  equation  of  the  given  curve. 

The  student  may  get  a  somewhat  clearer  idea  of  this  process  if 
he  will  consider  the  analogy  between  it  and  the  work  of  trans- 
lating from  one  language  into  another.  In  translation  from 
English  into  German  for  example,  the  student  must  have  not  only 
a  knowledge  of  the  German  words  equivalent  to  the  English  words 
in  the  passage  to  be  translated,  he  must  also  have  a  knowledge 
of  the  peculiar  forms,  the  idiomatic  constructions,  of  the  two 
languages.  Ordinarily  he  will  first  of  all  throw  the  English 
sentence  into  the  German  order  and  replace  the  English  idiom  by 
the  corresponding  German  idiom,  and  then  is  ready  for  the  actual 
work  of  translation.  Now  algebra  is  after  all  to  a  great  degree 
merely  a  language,  and  an  equation  is  a  sentence.  The  equation 
of  a  locus  is  the  statement  in  algebraic  language  of  the  conditions 


30 


Analytic  Geometry. 


under  which  the  tracing  point  moves,  and  the  deduction  of  sucli 
an  equation  is  merely  the  translation  of  the  ordinary  English 
description  of  those  conditions  into  algebraic  language.  For 
example  consider  the  circle  of  radius  2  centered  at  the  point 
(4,  3).  The  algebraic  idiom  requires  first  of  all  that  the  curve 
be  described  as  the  locus  of  a  moving  point,  and  we  accordingly 
throw  our  description  of  the  curve  into  the  new  form,  "A  variable 
point  moves  in  such  a  way  as  to  keep  its  distance  from  the  point 
(4,  3)  equal  to  2."  The  phrase,  ^'A  variable  point"  translates  at 
once  into,  "The  point  (x^  i/)",  and  if  we  knew  an  algebraic  equiv- 
alent for  "The  distance  from  {x,  y)  to  (4,  3)''  we  should  at  once 
equate  it  to  2  and  have  the  equation  of  the  locus. 

Since  the  definitions  of  the  simpler  curves  are  largely  stated 
in  terms  of  distance  and  direction,  it  will  be  wise  before  we  take 
up  the  work  proper  of  this  chapter  to  develop  some  fundamental 
formulae  which  will  enable  us  to  translate  questions  of  distance 
and  direction  into  algebraic  language. 


21. 
DISTANCE 
BETWEEN  TWO 
POINTS  IN  TERMS 
OF  THEIR  CO- 
ORDINATES. 


Consider  any  two  points  F^  and  P^  whose 
co-ordinates  are  {x^,  y^)  and  (x^,  y^).  Draw 
Pi  P2.  Draw  Pi  Qi  and  P^  Q^  parallel  to  OY 
and  PjP  parallel  to  OX.  Let  D  be  the  re- 
quired distance.  Then  we  have  by  direct 
application  of  our  geometry 


D  =  PrP,=  y  P,lt  -I-  P,R 

But  PJt  =  Q^Q^  =  a?2  —  a?i 

and  P2R  =  P2Qti  —  PiQ2 

=  2/2  —  2/1 
therefore 


D=^/{x,—x,y  +{y^  —  y^y 


22. 

CO-ORDINATES  OF 
A  POINT  WITH  A 
GIVEN    DISTANCE 
RATIO. 


Fig.  9. 


Given        two 

points    Pi    and 

P2,  to  find  the 

co-ordinates    of 

a  third  point  Pg  on  the  straight  line  joining 
Pi  and  P2  which  shall  have  with  respect  to  Pi  and  Pg  a  given  dis- 
tance ratio.     (See  paragraph  9.) 


Analytic  Geometry. 


31 


Let  the  co-ordinates  of  Pi,  P^,  P^, 

be  (a?i,  ?/i),  fe,  2/2),  (a^s;  y-i)  and  let  the 

numerical  value   of 


be    ^ 


The    distance  ratio  P3  with  respect 
to  Pi  and    P2   will   then   be    either 

'  according  as  the  point  P3 


t^'^-x 


is   within  or    Avithout   the    segment 

PrP,. 

Consider    the    case    where    Pg    is 
within  the  segment  P1P2 .     We  have 


Fig.  10. 


at  once 


therefore 


and 


P.P.- 

P,S  _ 

p.p^ 

P^P 

A  — ^ 

—  a", 

\       a?2 

—  a?3 

2  +  Kx, 

\-\-\ 


PROBLEMS. 


1.  Show  from  the  same  figure  that  1/3  =    '^('^       ^^^^ . 


\-V\ 


2.  Construct  the  figure  for  the  case  when  P3  is  without  the 
segment  PiPg?  and  show  that  the  same  formulae  hold. 

3.  In  the  work  of  both  paragraphs  21  and  22  the  axes  have  been 
rectangular.  Deduce  the  corresponding  formulae  when  the  axes 
are  oblique. 

4.  Find  the  distance  from  (2,  3)  to  (5,  —1)  ;  (4,  1)  ;  (—5,  1)  ; 
(-1,-1);  (0,3). 

5.  What  is  the  general  formula  for  the  distance  of  a  point  from 
the  origin  ? 

6.  Express  the  co-ordinates  of  the  middle  point  of  a  segment  of 
a  line  in  terms  of  the  co-ordinates  of  its  extremities. 

7.  Find  the  co-ordinates  of  the  points  which  divide  the  segment 
of  the  line  terminating  at  (1,  3)  and  (4,  —  2)  into  three  equal  parts 
and  find  the  length  of  these  parts. 


32  Analytic  Geometry. 

8.  Find  the  co-ordinates  of  the  points  which  have  the  given  dis- 
tance ratios  with  respect  to  the  following  pairs  of  points.  The 
shorter  segment  is  in  each  case  the  one  terminating  at  the  point 
first  named. 


Points. 
(1,  4)        (2,  3) 

Ratios. 
2 
3 

(3,  2)        (-1,  0) 

(2,-1)  (-2, -4) 
(5,  6)       (1,  —3) 

2 
2 
3 
4 

(0,  0)-       (-2,  5) 

7 

2 

23.  In  analytic  geometry  the  angle  which  a  line 
ANGLE  BETWEEN  makes  with  the  X  axis  is  always  measured 
A  LINE  AND  THE  from  the  positive  end  of  the  X  axis  toward 
^  AXIS.                         the  positive  end  of  the  Y  axis.    Remembering 

this,  the  student  should  have  no  difficulty  in 
showing  that  the  tangent  of  the  angle  made  with  the  X  axis  by  the 
line  through  two  points  is  given  by  the  formula, 

tan^  =  |^' 

where  0  is  the  angle  and  {x^,  i/J,  {x^,  y^)  are  the  points.  He 
should  also  show  that  this  formula  holds  for  all  possible  positions 
of  the  line. 

PROBLEM. 

Find  the  angles  which  the  lines  of  problem  8  of  the  last  para- 
graph make  with  the  X  axis. 

24.  Now  that  w^e  have  developed  our  formulae 
ILLUSTRATIONS  OF  for  distance,  distance  ratio,  and  direction  of 
THE  DEVELOPMENT  a  straight  line  we  are  ready  to  take  up  again 
OF  EQUATIONS  OF  the  problem  we  were  compelled  to  leave  un- 
^^^'-  finished  in  the  latter  part  of  paragraph  20. 
We  now  are  able  to  translate  the  phrase,  "The  distance  from  {x,y) 
to  (4,  3)"  by  the  expression  V(^  —  4)^  -\-{y  —  3)-,  and  the  state- 
ment of  the  way  in  which  the  variable  point  traces  the  curve  now 
translates  into 


V(a^  — 4)2+(i/-3)2  =  2 


Analytic  Geometry.  33 

which  is  the  equation  of  the  curve,  since  it  is  the  algebraic  state- 
ment of  the  necessary  and  sufficient  condition  that  the  point 
{x,y)  may  lie  on  the  circle. 

It  is  usual  to  reduce  such  expressions  however  to  the  simplest 
possible  form,  and  the  equation  of  this  circle  would  usually  be 
written  in  the  form 

x'  +  y^'  —  ix  —  2i/  — 11  =  0.* 

Let  us  consider  a  second  example  of  this  sort.    The  line  joining 

a  variable  point  to  the  point  (1,  2)  makes  with  the  X  axis  an  angle 

whose  tangent  at  any  instant  is  equal  to  the  abscissa  of  the  variable 

point  at  the  same  instant.    Find  the  locus  of  the  variable  point. 

"The  variable  point"  translates  into  (Xy  y).    ''The  abscissa  of  the 

variable  point"  translates  into  x.    "The  tangent  of  the  angle  made 

by  the  line  with  the  X  axis"  translates  (by  paragraph  23)  into 

V  —  2 

- — — -.  Therefore  the  algebraic  statement  of  the  condition  under 

X  —  1  * 

V  —  2 
which  the  curve  is  described  is  evidently      -  =  j? 

i.e.,  y  =  x'^  —  x-\-2. 

If  the  student  desires  to  know  the  form  of  the  curve  it  can  easily  be 
plotted.  Later  on  he  will  learn  to  classify  simple  curves  without 
plotting. 

PROBLEMS. 

1.  Find  the  locus  of  all  points  equally  distant  from  (1,  1)  and 
(2,  4)  ;  from  (1,  3)  and  (—1,  5). 

2.  Generalize  problem  1  by  taking  {x^,  ^/i)  ^^^  (^2?  2/2)  ^^  the 
two  fixed  points,  and  show  that  the  equation  is  always  of  the  first 
degree. 

3.  Find  the  equation  of  the  circle  whose  center  is  at  (a,!))  and 
whose  radius  is  r.  (Since  proper  choice  of  a,  1),  and  r  will  make 
this  any  circle  whatever,  the  corresponding  equation  is  called  the 
general  equation  of  the  circle.) 

4.  Generalize  the  second  illustration  of  this  paragraph  by  replac- 
ing the  point  (1,  2)  by  a  point  {x^,  yj. 

5.  A  point  moves  so  that  the  square  of  its  distance  from  (3,  2) 
plus  the  square  of  its  distance  from  (1,  3)  equals  27,  find  the 
equation  of  the  locus.  Does  a  comparison  of  the  result  with  that  of 
problem  3  give  any  hint  as  to  the  nature  of  the  curve  ? 

6.  Show  that  if  a  point  moves  so  that  the  sum  of  the  squares  of 
its  distances  from  three  fixed  points  is  constant,  the  equation  of 


'^See  appendix  C. 


34  Analytic  Geometry. 

its  path  will  always  be  of  the  second  degree,  will  have  no  term  in  xy^ 
and  will  have  the  same  coefficient  for  the  terms  in  a?-  and  y^.  Can 
these  statements  be  extended  to  a  greater  number  of  points? 

7.  Q^  and  Q2  are  two  fixed  points  and  P  is  a  variable  point.  The 
movement  of  P  is  subject  to  the  condition  that  the  tangents  of  the 
angles  which  the  two  lines  PQ^  and  PQo  make  with  the  X  axis  shall 
be  numerically  equal  but  of  opposite  sign.    Find  the  locus  of  P. 

8.  A  moving  point  traces  a  straight  line  passing  through  the 
point  (1,  —  2)  and  making  with  the  X  axis  an  angle  whose  tangent 
is  2.    Show  that  the  equation  of  the  line  is 

y  =  2x  —  4 

9.  A  line  has  an  intercept  on  the  Y  axis  of  4  (i.  e.,  passes  through 
the  point  (0,  4))  and  makes  with  the  X  axis  an  angle  whose 
tangent  is  3.    Show  that  its  equation  is 

y==Sx  +  4. 

10.  Since  any  line  may  be  defined  by  its  intercept  on  the  Y  axis 
and  the  angle  it  makes  with  the  X  axis,  generalize  problem  9  and 
show  that  the  equation  of  any  straight  line  is  of  the  first  degree 
and  of  the  form 

y  =  mx  +  Ti. 
What  are  m  and  7i? 

11.  Show  conversely  that  any  equation  of  the  first  degree  in  x 
and  y  can  be  reduced  to  the  form 

y  =  mx  +  h 

and  therefore  represents  a  straight  line. 

The  work  of  this  paragraph  will  be  resumed  after  the  student  has 
acquired  a  greater  amount  of  material  on  which  to  base  problems. 


CHAPTER  VII. 

TO  DEDUCE  THE  EQUATION  WHEN  A  FINITE  NUMBER 
OF  POINTS  ON  A  LOCUS  OF  SOME  KNOWN  TYPE  IS 

GIVEN. 

25.  If  we  are  given  merely  a  number  of  points 

LIMITATIONS  OF  on  a  locus  there  is  no  way  by  which  the 
THE  PROBLEM.  equation  may  be  deduced,  but  if  in  addition 

OUTLINE  OF  THE  fo  a  number  of  points  we  are  given  such  ad- 
METHOD.  ditional  information  as  will  enable  us  to 

determine  the  general  form  of  the  equation,  the  problem  is  at  once 
simplified.  For  if  the  form  of  the  equation  is  known,  each  point 
that  satisfies  the  equation  gives  us  a  relation  connecting  the 
coefficients;  and  the  complete  determination  of  the  equation  is 
therefore  possible  whenever  a  sufficient  number  of  points  have  been 
given.  For  example,  let  it  be  known  that  a  straight  line  passes 
through  (4,  7)  and  (3,  5).  Problem  10  of  the  last  paragraph  tells 
us  that  the  equation  must  be  of  the  first  degree  and  therefore  of  the 
general  form 

Now  if  the  two  points  lie  on  the  locus  their  co-ordinates  must 
satisfy  the  equation  and  we  have 


4 

A 

C 

+  7 

f  +  - 

=  0 

3 

A 

C 

+  5 

f  +  - 

=  0 

A_ 

C~ 

=  - 

-2- 

and    |: 

=  1 

whence 

and  the  equation  of  the  line  is 

—  2x  +  y-\-l  =  {i. 


*This  form  contains  apparently  three  arbitrary  constants,  but  division 
by  any  one  of  them  reduces  the  equation  to  a  form  which  contains  only  two. 
Such  an  equation  is  said  to  contain  two  effective  constants. 


36  Analytic  Geometry. 

Again,  problem  3  of  the  last  paragraph  shows  us  that  the  equa- 
tion of  the  circle  whose  center  is  at  {a^  J))  and  whose  radius  is  r 
is  of  the  form 

This  equation  contains  three  effective  constants,  but  it  is  of  the 
second  degree  in  these  constants,  and  our  subsequent  work  will 
therefore  gain  in  simplicity  if  we  reduce  the  equation  to  the  form 

ar.2  _|_  2/2 _ 2ax—2hij  +  a^  +  &2 _ r^  =0, 
or,  putting     a^-\-'b'^  —  r^  =  —  c, 

^'  +  2/'  — 2rta?  — 2&2/  — c  =  0. 

If  now  the  three  points  (1,  1),  (3,  3),  (4,  1)  lie  on  the  curve  we 
have 

2a  +  26  +  c  =  2 
6a  +  6&  +  c  =  18 

8a  +  2&  +  c  =  17 

whence  the  student  may  find  a,  h,  c  and  so  determine  the  equa- 
tion of  the  circle. 

26.  .  The   fact  that   the   co-ordinates   of  every 

NUMBER  OF  POINTS  point  on  a  curve  must  satisfy  the  equation 
REQUIRED  TO  of  the  curve,  taken  in  connection  with  the 

DETERMINE  A  algebraic  theorem  that  n  non-homogeneous 

CURVE.  equations  are  necessary  and  sufficient  to  de- 

termine n  unknown  quantities,  leads  us  at  once  to  the  important 
theorem  : 

The  number  of  effective  arbitrary  constants  in  an  equation  \a 
equal  to  the  number  of  arbitrary  points  through  which  the  corre- 
sponding curve  may  be  made  to  pass. 

PROBLEMS. 

1.  Find  the  equations  of  the  straight  lines  through  the  follow- 
ing pairs  of  points : 

(1,3)  (2,-1);  (2,4)  (3,0);  (4,  3)  (2,  4)  ;  (4,  3)  (-4,-3). 

A  B 

Jn  the  last  case  the  values  of  —and  —  are  infinite.    This  of 

course  means  merely  that  G  is  zero.  The  difficulty  may  be  avoided 
by  dividing  by  A  in  place  of  C. 

2.  Find  the  equation  of  the  circle  through  the  points  (1,  2), 
(2,4),  (1,4). 

3.  A  curve  of  the  form  y^  =  2px  passes  through  the  point 
(4,  2) .    Determine  the  value  of  p. 


Analytic  Geometry.  37 

4.  Show  that  the  equation  of  any  straight  line  through  the 
origin  is  of  the  form  y  =  mx  where  m  is  an  arbitrary  constant. 
What  is  the  geometric  significance  of  m  ? 

5.  Find  the  equation  of  the  circle  through  the  three  points 
(—1,2),  (-1,-3),  (0,0). 

6.  Show  that  the  equation  of  any  curve  which  passes  through 
the  origin  can  have  no  constant  term. 

7.  How  many  effective  constants  are  there  in  each  of  the  follow- 
ing equations? 

ax''  +  hif  +  21ixij  +  2(/a?  +  2f^  4-  c  =  0 
{ax-{--by-{-c){dx  +  fy  +  g)={)' 
'  a{l)x  -\-  cy  -{-  d)=^ 

(a  -\-  i)) X  -\-  cy ^=^ 
ax^  +  'by  —  ca?  +  2  =  0 

27.  In  place  of  giving  points  on  the  curve,  some 

COMPLEX  other  condition  may  be  stated  which  is  equiv- 

CONDITIONS  1      ^  X        •    •  '  .    X         T^ 

EQUIVALENT   TO        ^lent  to  giving  one  or  more  points.    For  ex- 
TWO  OR  MORE  ample,  to  give  the  center  of  a  circle  is  equiv- 

SIMPLE  ONES.  alent  to  giving  both  a  and  1),  and  therefore 

is  equivalent  to  giving  two  points  on  the 
curve.  Again,  to  give  the  angle  w^hich  a  line  makes  with  the 
X  axis  is  to  determine  m,  and  therefore  is  equivalent  to  giving 
one  point. 

PROBLEMS. 

1.  Find  the  equation  of  the  circle  which  passes  through  (3,  —  1) 
and  has  its  center  at  (4,  2) . 

2.  Find  the  equation  of  the  circle  which  passes  through  (3,  —  1) 
and  (1.  4)  and  has  its  center  on  the  X  axis. 

3.  Find  the  equation  of  the  straight  line  through  the  point 
(3,  4) ,  making  an  angle  with  the  X  axis  of  70°,  110^  45°,  135°. 

4.  A  given  line  makes  with  the  X  axis  an  angle  whose  tangent 
is  m,  and  has  an  intercept  a  on  the  X  axis.  Show  that  its  equation 
is 

y  =  m(x  —  a). 

5.  Find  the  equation  of  a  line  through  (1,  5),  making  an  angle 
of  45°  with  the  X  axis. 

0.  Find  the  equations  of  the  lines  through  ( — 2,  5)  parallel  to 
the  lines  of  problem  3. 

7.  Find  the  equation  of  the  line  through  the  point  ( —  3,  —  4) 
parallel  to  the  X  axis,  to  the  Y  axis. 


CHAPTER  VIII. 
TRANSFORMATION  OF  CO-ORDINATES/ 


28.  If  the  student  turns  back  to  paragraph  10, 
THE  GENERAL  he  will  note  that  our  present  system  of  co- 
problem,                    ordinates  rests  on  certain  assumptions  which 

are  equivalent  to  the  arbitrary  determination 
of  the  following:  the  origin,  the  direction  of  one  axis, 
the  angle  between  the  axes,  the  scales  of  measurement. 
The  exigencies  of  the  discussion  may  at  any  time  demand  a  change 
in  any  one  of  these,  and  the  algebraic  significance  of  such  a 
change  must  therefore  be  investigated.  As  in  the  case  of  our 
work  in  one  dimension,  we  desire  to  find  formulae  which  will  give 
us  the  old  variables  in  term-S  of  the  new.  We  consider  in  turn 
the  formulae  corresponding  to  changes  in  each  of  the  four  as- 
sumptions mentioned  above. 

29.  To  change  the  first  assumption  without 
movement  of  the  producing  any  change  in  any  of  the  others 
AXES  PARALLEL  TO  it  is  sufficient  to  move  the  axes  parallel  to 
themselves.             themselves.    Let  XOY  be  the  original  system 

and  by  such  a  movement  secure  a  second 
system  X'0'Y\  where  the  co-ordinates 
of  0'  referred  to  XOY  are  a  and  h. 
Let  P  be  any  point  in  the  plane  and 
let  the  co-ordinates  of  P  be  (x,  y)  in 
the  first  system  and  (a?',  y)  in  the 
second.    Then  for  all  positions  of  P 

we  have 

x  =  x  -\-  a 

y=y  +fy 

which  are  accordingly  the  formulae 
of  transformation. 


> 

r         ^ 

? 

6 

0 

Fi 

a.  11. 

•The  subdivision  of  our  subject  adopted  in  paragraph  12  seems  to  call  at 
this  point  for  a  discussion  of  the  last  of  the  problems  there  stated,  but  the 
discussion  is  in  many  cases  so  facilitated  by  transforming  the  axes  that  it 
seems  wise  to  introduce  the  present  chapter  at  this  point.  The  student  who 
is  not  thoroughly  familiar  with  the  simpler  theorems  concerning  the  pro- 
jection of  plane  contours  should  read  Appendix  D  before  undertaking  the 
work  of  this  chapter. 


Analytic  Geometry.  ^  39 

PROBLEMS. 

1.  Find  the  formulae  corresponding  to  a  change  to  a  new 
set  of  axes  parallel  to  the  old,  but  with  the  new  origin  at  the 

point  (1,  — l;,  (4,0),  CO,  4),  (-2— 8),  (7,1)- 

2.  Find  the  formulae  of  transformation  corresponding  to  the 
following  movements  of  the  axes : 

The  X  axis  1  upward,  Y  axis  4  to  the  right; 
The  X  axis  4  downward,  Y  axis  unmoved; 
The  X  axis  unmoved,  Y  axis  %  to  the  left. 

3.  What  movements  of  the  axes  correspond  to  the  following 
substitutions? 

^a?  =  a?'  +  4  (a?  =  a?'  —  2 

\y=y—^  \y=y\ 

\y=y'—^  \y  =  y+^ 

4.  In  the  figure  the  axes  are  rectangular.    Will  the  same  form- 
ulae hold  in  case  the  axes  are  oblique? 

5.  Bj  a  movement  of  the  X  axis  reduce  the  equation 

2/=r3a?  +  4 

to  an  equation  in  w'  and  y'  which  has  no  constant  term.  (Assume 
the  axis  to  be  moved  a  distance  a,  make  the  substitution,  and  then 
determine  a  by  equating  the  constant  term  in  the  transformed 
equation  to  zero.) 

6.  Do  the  same  thing  by  a  movement  of  the  Y  axis. 

7.  Free  the  equation 

«^'  +  2/'  — 2^  — %  +  l  =  0 

from  the  terms  of  the  first  degree  in  a?  and  i/  by  a  movement  of  the 
origin,  keeping  the  axes  parallel  to  their  original  position. 

8.  Move  the  origin  so  that  the  equation 

W  +  '^fy  +  ^g^  +  G  =  ^ 

shall  be  transformed  to  an  equation  having  no  term,  of  the  first 
degree  in  y  and  no  constant  term.  Is  such  a  movement  always 
possible? 

9.  What  must  be  the  co-ordinates  of  the  new  origin  in  order 
that  the  most  general  equation  of  the  second  degree 

ffa?2 +bi/2+27ia?t/  +  2^^  +  2fz/+c  =  0    . 


40 


Analytic  Geometry. 


may  reduce  to  an  equation  which  has  no  terms  of  the  first  degree 
in  the  variables  ?*    Is  the  operation  possible  when 

h^  =  aJ)f 

10.  Are  there  any  terms  of  an  equation  which  cannot  be  removed 
by  a  transformation  of  the  kind  we  have  been  considering? 


30. 

ROTATION   OF  THE 
AXES. 


To  change  the  second  assumption  without 
affecting  any  of  the  others  it  is  sufficient  to 
rotate  the  axes  about  the  origin  into  a  new 
position  in  which  they  make  an  angle  0  with 


the  old  position. 

Let  XOY  be  the  axes  in  the  first 
position  and  X'O'Y'  the  axes  in 
the  new  position.  Let  P  be  any 
point  having  the  co-ordinates  {x,  H^ 
y)  and  {x,  y)  in  the  two  systems. 
Then 

Oa  =  X,  P.a  =  y, 

Oc  =  x\  Pc  =  y. 
The  two  contours  OaP  and  OcP 

have  the  same  terminal  points  and  \  fig.  12. 

their  projections  on  any  line  are  therefore  equal.    Hence  we  have, 
by  projecting  on  OX, 

Oa  cos  0  +  aP  cos  —  ="  Oc  cos  0  +  cP  cos  (0  -\-  — ) . 

Similarly  by  projecting  the  same  contours  on  the  line  OY  we  have 

Oa  cos  —  +  aP  cos  0  =  (9c  cos   (  ^  —  0\-\-  cP  cos  0, 

and  these  two  equations  at  once  reduce  to 

.X  =  X  cos  0  —  y  sin  0 
y  =  X  sin  0  +  //  cos  0 

which  are  the  desired  formulae  of  transformation. 


*If  an  equation  of  the  second  degree  has  no  terms  of  the  first  degree  in 
the  variables  it  is  of  the  form 

aa?2  -f  hy2  _|_  2hxy  -f  c  -=  0. 
If  such  an  equation  is  satisfied  by  a  point  {x^,  y^)  it  will  "also  be  satisfied 
by  the  point  ( — x  ,  — y  ).  The  locus  is  therefore  symmetrical  with  respect 
to  the  origin,  and  our  problem  might  be  thus  stated  :  To  move  the  origin 
to  the  center  of  symmetry  of  the  curve  represented  by  the  equation  of  the 
second  degree. 


Analytic  Geometry.  41 

If  we  desire  the  formulae  which  correspond  to  a  change 
from  X'OY'  to  XOY  we  may  solve  these  equations  for  a)'  and 
y',  or  we  may  project  on  the  lines  OX'  and  OY' ,  or  we  may  note 
that  this  new  case  differs  from  the  original  one  only  in  the  angle, 
which  is  now  negative.    Any  one  of  these  methods  will  give  us 

x'  =^  X  cos  0  -\-  y  sin  6 
y'  =  —  X  sin  6  -\-  y  cos  0 

PROBLEMS. 

1.  Write  the  formulae  of  transformation  which  correspond  to 
the  following  rotations,  putting  in  in  each  case  the  numerical 
values  of  cos  0  and  sin  0 

60°,  30°,  45°,  —30°,  150°,  (tt— 60°),  (f  ^— 30°V  ^  ,  tt. 

2.  Show  that  the  equation 

X-  —  xy  —  2=0 

may  be  freed  from  the  term  in  xy  by  a  proper  rotation  of  the  axes. 
(Apply  the  proper  formulae  and  the  equation  becomes 

x"  (cos^  0  +  cos  0  sin  0)  +  y"  { sm^  6>  —  cos  (9  sin  0) 

+  x'lj  (cos'  0  —  sin'  0—2  sin  0 cos  (9)  —  2  ==  0. 

In  order  that  there  may  be  no  term  in  x'y   we  must  so  choose 
^that    - 

cos'  0  —  sin'  0  —  2  sin  0  cos  ^  =  0 
or  cos  20  —  sin  20  =  {) 

tan  2(9  ==  1 

(9=  22°  30'  =  ^. 
8 

3.  Free  the  equation 

x"^  —  xy  -\~Sy  —  2^  =  0 

from  the  term  in  y  by  a  rotation  of  the  axes. 

4.  How  many  terms  may  be  removed  from  an  equation  by  a  rota- 
tion of  the  axes?  Are  there  any  terms  which  are  unaffected  by 
such  a  rotation? 

5.  Through  what  angles  may  the  axes  be  turned  without  intro- 
ducing an  xy  term  into  the  equation 

6.  Show  that  the  general  equation  of  the  second  degree  will 

reduce  to  an  equation  without  an  xy  term  if  the  axes  are  rotated 

2/i 
through  an  angle  0  such  that  tan  20  =  ^    '^         . 

yd  0) 


42 


Analytic  Geometry. 


31. 


CHANGE  OF  THE 
ANGLE  BETWEEN 
THE  AXES. 


To  change  the  third  assumption  without 
affecting  any  of  the  others  it  is  sufficient  to 
change  the  direction  of  either  of  the  axes, 
but  it  will  be  better  to  develop  the  more  gen- 
eral formulae  corresponding  to  a  change  of 
the  directions  of  both  axes.  ^ 

Let  XOY  be  the  first  set  of  axes 
and  TOY'  the  second.  L^t  OX'  make 
an  angle  a  and  OY'  an  angle  /3  with 
OX^  and  let  the  angle  between  OX' 
and  or  be  8  (  8=  i^  —  a).  Let  P 
be  any  point  (x^  y)  in  the  plane. 
Then  by  considering  the  projections 
of  OaPawdi  OcP  on  OX  and  OYy^e 
have 

X  =  X  cos  a  +  \i  cos  ^ 
y  =  x  sin  cl-\-  y'  sin  P, 

a  set  of  formulae  which  are  sufficient  to  transform  from  any  set 
of  rectangular  axes  to  any  set  of  oblique  axes  which  have  the  same 
origin. 

If  we  wish  to  transform  from  oblique  to  rectangular  axes  we 
have  only  to  solve  the  formulae  just  derived  for  x  and  y  and  we 
have 

/      X  sin  P  —  ?/  cos  P 

^  = .    i 

sin  0 
t — X  sin  g  +  j/  cos  a 


Fig.  13. 


1.  Free  the  equation 


sin 


PROBLEMS. 


Ja?— 3?/  +  l=0 


from  its  term  in  7/  by  a  change  in  the  direction  of  the  Y  axis. 
2.  What  form  does  the  equation 


a?2  -|-  2/^  =  r^ 

take  when  the  X  axis  is  left  unchanged  but  the  Y  axis  moved  so 
that  the  angle  between  the  axes  is  30°  ? 


Analytic  Geometry.  43 

32.  To  change  the  last  of  our  assumptions 
CHANGE  OF  THE  without  affecting  any  of  the  others  it  is 
SCALES  OF                 sufficient  to  write 

MEASUREMENT. 

X  =  Kx 
y  =  Ly\ 

The  effect  is  evidently  to  multiply  the  unit  of  measurement  on 
the  X  axis  by  K  and  on  the  Y  axis  by  L. 

33.  The  transformation  from  any  system  of 
THE  GENERAL  axes  to  any  other  may  now  be  accomplished 
TRANSFORMATION,    by  the  proper  combination  of  the  formulae 

developed  in  the  preceding  paragraphs. 

PROBLEMS. 

1.  Write  the  formulae  of  transformation  which  correspond  to 
a  movement  of  the  origin  to  the  point  (4,  5)  and  a  rotation  of 
the  axes  through  60°. 

2.  Write  the  formulae  w^hich  correspond  to  the  transformation 
from  rectangular  axes  to  a  new  set  of  oblique  axes,  whose  origin 
is  at  ( —  4,  3)  and  whose  X  and  Y  axes  make  angles  of  30°  and  95° 
with  the  original  X  axis. 

3.  The  origin  is  moved  to  the  point  ( — 2,  — 4),  and  the  axes 
are  rotated  through  an  angle  of  10°.  Write  the  formulae  of 
transformation. 

4.  Show  that  none  of  the  transformations  so  far  discussed  can 
change  the  degree  of  an  equation.  (It  is  sufficient  to  show  that  it 
cannot  be  raised.  For  if  a  change  of  the  axes  transforms  an 
equation  into  one  of  lower  degree,  change  the  axes  back  to  their 
original  position  and  the  equation  will  be  restored  to  its  original 
form,  i.  e.,  the  degree  of  the  transformed  equation  will  be  raised. 
Therefore  when  the  student  has  shown  that  the  degree  of  an 
equation  cannot  be  raised  he  has  shown  that  it  cannot  be  lowered.) 

34.  Each  of  the  above  transformations  has 
TRANSFORMATIONS  been  interpreted  as  corresponding  to  a 
INTERPRETED  AS  change  in  the  system  of  co-ordinates.  There 
CHANGES  OF  THE  is  however  another  interpretation  which  is 
LOCI.  frequently  adopted.  Consider  any  equation 
as  referred  to  a  given  system  of  reference.  Apply  any  one  of  the 
above  substitutions.  The  result  will  be  a  new  equation,  w^hich 
may  of  course  be  referred  to  the  original  system  of  reference  and 


44  Analytic  Geometry. 

when  so  referred  represents  a  new  locus.  The  transformation 
may  thus  be  regarded  as  a  change  in  the  locus  instead  of  a  change 
in  the  system  of  reference.    For  example,  apply  to  the  circle 

the  substitution 

x=x'-\-a         y=y  -\-^ 

and  refer  the  resulting  equation 

to  the  original  set  of  axes.  From  this  point  of  view  the  trans- 
formation has  evidently  resulted  in  moving  the  center  of  the  circle 
from  the  point  {a,  h)  to  the  origin.  The  student  will  find  it  in- 
teresting to  study  all  of  the  above  transformations  from  this 
second  point  of  view. 


CHAPTER  IX. 

INTERSECTION  OF  LOCI. 

35.  We  have  found  that  any  equation  connect- 

THE  SIGNIFICANCE  ing  x  and  y  represents  a  locus  every  point  of 
OF  IM  AGIN  ARIES.  Avhich  satisfies  the  equation.  The  following 
question,  which  was  given  a  somewhat  super- 
ficial treatment  in  paragraph  11,  now  demands  more  careful 
consideration.  ^'' Given  two  equations,  is  it  possible  to  find 
a  point  or  points  wiiich  lie  on  both  loci  and  therefore 
satisfy  both  equations?"  Looked  at  from  the  geometric  side 
the  question  is,  "Do  the  loci  intersect?''  and  the  answer  is,  "They 
may  or  may  not  according  to  their  relative  positions."  Looked  at 
from  the  algebraic  side  the  question  is,  "Can  two  equations  in  two 
variables  be  simultaneously  satisfied?"  and  the  answer  is,  "Yes, 
without  exception."  The  cause  of  this  apparent  discrepancy'  lies 
in  the  nature  of  our  fundamental  assumptions,  which  were  so 
made  as  to  establish  a  one  to  one  correspondence  between  the 
points  of  the  plane  and  pairs  of  real  values  of  x  and  y,  while  the 
theorem  that  two  equations  in  two  variables  can  always  be  simul- 
taneously satisfied  holds  true  only  Avhen  pairs  of  imaginary  values 
are  included. 

To  make  the  matter  a  little  clearer  consider  the  equation 

x  —  y  =  {). 

Pairs  of  values  that  will  satisfy  this  equation  are  of  two  kinds; 
real  values  such  as  x  =  a,  y  =  a,  or  complex  values  such  as 
x=^a  -[-  ih,  y  =  a  -\-  iJ),  including  pure  imaginaries  x  =  ih,  y  =  ih. 
The  first  kind  alone  corresponds  to  points  in  the  plane  and  includes 
all  the  points  on  the  line  through  the  origin  bisecting  the  angles 
in  the  first  and  third  quadrants.  It  is  evident  that  there  is 
not  a  complete  correspondence  between  this  line  and  the  equation 
X —  y  =  0,  since  the  equation  has  a  much  more  general  significance 
than  the  line;  and  it  is  also  evident  that  this  lack  of  complete 
correspondence  is  due  to  the  nature  of  our  fundamental  assump- 
tions, which  give  us  no  geometric  representation  for  pairs  of 
imaginary  values. 


46 


Analytic  Geometry. 


It  is  however  frequently  desirable  to  be  able  to  state  algebraic 
theorems  in  geometric  language,  and  so  mathematicians  are  ac- 
customed to  speak  of  these  pairs  of  complex  values  of  x  and  y  as 
representing  imaginary  points.  From  this  point  of  view  the  curve 
corresponding  to  any  equation  f{x,y)=^{)\^  considered  to  consist 
of:  («)  an  infinity  of  real  points  which  constitute  the  visible  curve : 
(6)  an  infinity  of  imaginary  points  just  as  intimately  associated 
with  the  equation,  but  having  no  representation  in  the  diagram. 
(See  appendix  E.) 

Any  point,  real  or  imaginary,  which  belongs  to  each  of  two 
curves  is  called  an  intersection,  real  or  imaginary,  of  the  curves. 
From  the  algebraic  theorem  that  two  equations  of  degree  m  and  r? 
in  two  variables  can  always  be  satisfied  by  mn  pairs  of  values  of 
the  variables  it  follows  that  two  curves  of  degree  m  and  n  intersect 
in  mn  real  or  imaginary  points,  some  of  which  may  in  special 
cases  coincide  with  each  other. 
36. 


POSSIBILITY  OF 
ERROR. 

and 


So  long  as  both  the  equations  are  of  the 
first  degree  no  ambiguity  in  the  results  \^ 
possible.     For  example,  the  two  equations 

2^  —  1/  —  3  =  0 
4:X  —  y  —  7  =  0 

yield  on  solution  the  two  equations 

x  =  2         y  =  l 

showing  that  the  intersection  is  on  the  line  parallel  to  the  Y  axis 
at  a  distance  2  to  the  right,  and  on  the  line  parallel  to  the  X  axis 
at  a  distance  1  above. 

But  consider  the  intersections  of  Y 

the  circle  of  Fig.  14,  whose  equation 
is 

2a?2  +  21/-  =  3 

and  the  curve  A'B'C'D'  whose  equa- 
tion is 

^'  +  ^■^  +  2/^  =  2. 

Solving  these  two  equations  for  x 
we  have 


r  20 


Fig.  14. 


Analytic  Geometry.  47 

showing  that  the  points  of  intersection  are  on  one  of  the  four 
lines  AA',  BB',  CC\  DD\    If  now 


4 


1    20 


for  example,  be  substituted  for  x  in  the  equation  of  the 
circle  we  obtain,  on  solving  the  resulting  equation  in  //,  the  y 
co-ordinate  of  the  intersection  A^  but  we  also  obtain  the 
y  co-ordinate  of  the  point  A''  in  which  we  have  no  interest; 
wiiile  if  the  same  value  of  x  is  substituted  in  the  equation 
of  the  other  curve,  we  obtain  the  y  co-ordinates  of  the  points 
A  and  A\  In  such  cases  as  this  the  student  must  determine 
by  substitution  in  the  equations  w^hich  of  the  points  obtained 
correspond  to  the  actual  intersections  of  the  curves. 

PROBLEMS. 
Find  the  intersections  of  the  following  pairs  of  curves : 

1.  2x  +  By  +  l  =  0  2.  w  —  y  =  0 

4a?  —  ■?/  +  2  =  0 
3.  a?  —  2i/  +  1  =  0 

—  4a?  +  2^  —  7  = 
5.  a?  — 2ii/  +  3  =  0 

a?  +  2ii/  +  3  =  0 
7.  £p2  _|_  I/'  +  4  =  0 

^  +  22/  +  l  =  0  ' 
9.  4^2  _j_  3^2  _  12  = 

3a?2  —  iy^  —  12  = 

11.  Find  the  intersections  of 

y  =  a 
with  ^2  _p  2/'  =  ^^ 

and  stnte  in  both  algebraic  and  geometric  terms  what  happens 
as  a,  at  first  less  than  6,  gradually  increases  till  it  is  greater  than 
h. 

12.  Find  the  intersection  of 

y  =  m,x  +  h^ 
and  y  =  m2X-\-'b2 

and  di«cnss  the  case  when  m^  =  m^. 

Find  the  intersections  of  the  following  curves  with  the  axes : 

13.  y  =  ma-  +  &  14.    _^  _|-  J/  ^  1 

a        h 

15.  Aa?  +  ^i/-f  (7  =  0  16.  x^  —  2ay  +  l  =  ^ 


x-\-y  =  {) 

4.  x  —  iy-\-2  =  ^ 

0 

a?  +  ii/  +  4  =  0 

6.  ^2_|_^2^4 

a?  +  2i/  =  0 

8.  a?2  +  2/'— 2  =  0 

a?  +  2/  — 4  =  0 

0 

10.  4a?2  _|_  9^2  ^  30 

0 

9a?2_|_4^2^3g 

CHAPTER  X. 

THE  EQUATION  OF  THE  FIRST  DEGREE  AND  THE 
STRAIGHT  LINE.* 

37.  Our  investigation  of  the  geometric  prop- 

STANDARD  FORMS  erties  and  relations  of  loci  by  means  of  their 
OF  THE  EQUATION  equations  will  be  facilitated  by  adopting 
OF  THE  STRAIGHT  some  mode  of  classification  of  equations. 
LINE.  That  by  degrees  is  probably  the  most  natural 

and  for  our  present  purpose  the  most  convenient. 

The  student  has  already  deduced  for  himself  a  number  of  im- 
portant results  concerning  the  equation  of  the  first  degree  and  its 
corresponding  locus  which  are  here  summarized  for  convenience 
of  reference. 

I.  Every  equation  of  the  first  degree  represents  a  straight  line. 
(Problem  11.    Paragraph  24.) 

II.  Conversely,  every  straight  line  is  represented  by  an  equa- 
tion of  the  first  degree.     (Problem  10.    Paragraph  24.) 

III.  If  the  equation  be  of  the  form 

y^mx  -\-  h 

m  is  the  tangent  of  the  angle  made  by  the  line  with  the  X  axis 
and  h  is  the  intercept  on  the  Y  axis.  (Problem  10.  Paragraph 
24.) 

IV.  If  the  equation  be  of  the  form 

y  =  m{x  —  a), 

m  has  the  same  meaning  as  before,  but  a  is  now  the  intercept  of 
the  line  on  the  X  axis.    (Problem  4.    Paragraph  27.) 


*We  begin  at  this  point  the  study  of  the  last  of  the  five  problems  men- 
tioned in  paranjraph  12  and  shall  continue  it  for  several  chapters.  The 
student  must  remember,  however,  that  any  such  division  of  a  subject  as  is 
attempted  in  stating  these  five  heads  is  of  necessity  somewhat  arbitrary,  and 
he  must  therefore  not  be  surprised  to  find  under  this  last  head  problems 
and  theorems  that  might  with  entire  propriety  be  stated  under  some  other 
heading.  In  particular  he  will  find  an  entire  chapter  (Chapter  XII)  devoted 
to  loci  problems  which  have  been  deferred  to  this  later  position  because  the 
student  had  not  at  an  earlier  date  sufficient  material  on  which  to  base  them. 


Analytic  Geometry.  49 

V.     If  the  equation  be  of  the  form 

—  +^=  1 
a        h 

a  and  h  are  the  intercepts  on  the  X  and  Y  axes.     (Problem  14. 
Paragraph  3t).) 

The  tangent  of  the  angle  made  by  the  line  with  the  X  axis  is 
frequently  called  the  slope  of  the  line,  and  the  forms  in  III  and  IV 
are  consequently  called  the  slope  equations  of  the  straight  line, 
while  the  form  in  V  is  called  the  intercept  equation  of  the  straight 
line.  Ax -\- By -\- C  =^^  is  called  the  general  equation  of  the 
straight  line. 

PROBLEMS. 

1.  Find  the  slopes  and  intercepts  of  the  following  lines : 

2a?  +  3?/  +  l  =  0  y  =  ix  +  2 

X  —  2i/  =  0  x  =  a 

2.  Compare  the  general  equation  with  the  slope  and  intercept 
equations  and  deduce  the  following  results  for  the  general  equation 

Slope  =- J 

C 
Intercept  on  the  X  axis  = ^ 

C 
Intercept  on  the  Y  axis  = 

3.  A  knowledge  of  the  intercepts  makes  the  plotting  of  the  line 
an  easy  matter,  excepting  in  one  special  case.  (What  is  this  ex- 
ception?)    Plot  the  lines  of  problem  1. 

4.  Since  the  slope  of  a  line  depends  on  the  ratio  —  what  con- 
clusion may  be  drawn  as  to  the  equations  of  parallel  lines  ? 

5.  Let  the  lines  AB  and  CD  have  the  equations 

y  =  m^x-^\ 
and  y^m^x+h^ 

respectively.    Show  that 

tan  ^  ='"■-'»' 


1  +  nil  m^ 
where  d  is  the  angle  between  the  lines. 


50  Analytic  Geometry. 

6.  Find  the  angles  between  the  following  pairs  of  lines : 

w  +  2y  —  l  =  0  2a?  —  21/ +  1  =  0 

a?  — 3i/  +  4  =  0  x  =  3y-{-2 

4x-i-y  —  2  =  0  2x  —  2y  +  l  =  0 

y==Sx  —  4:  a?  — 6//  +  4=0 

xr\-y  — 2  =  0  a?  — 2i/  +  3  =  0 

4a?  +  3  =  — 41/  y  +  2x-{-5  =  0 

7.  Show  that  the  angle  between  the  two  lines 

and  A^x  -{-B^y  -{-  €2  =  0 

is  given  by 

Ai  A2  ~T  B\  B-i 

8.  From  the  results  of  problem  5  determine  what  conditions 
must  be  satisfied  by  the  coefficients  of 

y^m^x  -\-  hj 
and  y  =  ni^x  -\-  h^ 

in  order  that  the  lines  may  be  parallel.    Perpendicular. 

9.  Determine  the  corresponding  conditions  for 

•       A,x  +  B,y^C,  =  0 
and  A^x  -^B^y  -{-€2  =  0 

10.  Remembering  that 

.  tan^  1 

sm  u  =         ,- ; — ,-^      and      cos  c^  = 


±  V  1  +  tan=^  e  ±  1/ 1  +  tan'  0 

deduce  formulae  for  the  sine  and  cosine  of  the  angle  which  a 
line  makes  with  the  X  axis  in  terms  of  m  and  U.  In  terms  of  A 
and  B. 

11.  A  line  is  subject  to  the  condition  that  it  must  be  parallel 
to  the  line 

2a?  — 3i/  +  l  =  0. 

To  what  extent  are  its  coefficients  determined  and  to  what  extent 
are  they  still  arbitrary? 

12.  Find  the  equations  of  lines  through  the  point  (2,  3)  parallel 
and  perpendicular  to  each  of  the  following  lines: 

^_2i/  +  3  =  0  0?  — 27/  +  3  =  0 

a?  =  3i/  -j-  4  a?  =  0 

—  i/-j-2a?  =  9  X  —  2/  =  0 

a?  ==  4i/  aa?  —  ci/  +  f  =  0      ^ 


Analytic  Geometry.  51 

13.  Are  there  any  eases  of  parallelism  or  perpendicularity 
among  the  following  lines? 

a?  —  2?/ +  3  =  0  a?  +  4i/  +  12  =  0 

i/  =  4iP  +  5  6i/— 3^  +  1  =  0 

7a?  — 2?/ +  3  =  0  ?/  — 4j?+3  =  0 

14.  A  line  is  subject  to  the  condition  of  passing  through  the 
point  (1,  3).  To  what  extent  are  its  coefficients  still  unde- 
termined ? 

15.  Find  by  the  method  of  paragraph  25  the  general  equation 
of  a  straight  line  through  the  point  (a?i,  y^). 

38.  If  we  attempt  to  find  the  equation  of  the 

THE  EQUATION  OF  straight  line  through  the  two  points  (co^,  y,) 
THE  STRAIGHT  LINE'  and  {w^,  y^) ,  the  method  of  paragraph  25  is 
THROUGH  TWO  of  course  perfectly  rigorous,  but  it  has  the 

GIVEN  POINTS.  disadvantage  of  demanding  an   amount  of 

algebraic  work  which  is  somewhat  wearisome,  and  in  certain  sim- 
ilar but  somewhat  more  complex  investigations  becomes  well  nigh 
prohibitive.  It  is  sometimes  possible  in  such  cases  to  avoid  much 
of  this  algebraic  work  and  infer  the  form  of  the  equation  desired 
from  principles  already  established.  Consider  for  example  prob- 
lem 15  of  the  last  paragraph.  The  equation  desired  must  be  of  the 
first  degree,  must  be  satisfied  by  the  point  (a?i,  i/i),  and  must  be 
sufficiently  general  in  its  form  so  that  we  may  be  able  to  satisfy 
it  by  the  co-ordinates  of  any  other  one  point  in  the  plane.  The 
equation  must  of  course  not  be  identically  zero.  Any  equation, 
no  matter  how  secured,  that  possesses  these  properties  is  the 
equation  desired.  Now  it  is  easy  enough  to  manufacture  an 
equation  which  possesses  them. 

y  —  y^  =  m{x  —  x^) 

in  fact  possesses  all  these  properties  and  will  hereafter  be  used 
as  the  equation  of  any  line  through  {x^,  y^).  (What  is  the  geo- 
metric significance  of  m?) 

The  problem  stated  at  the  heading  of  this  paragraph  may  be 
treated  in  the  same  way.  Here  the  equation  must  be  of  the  first 
degree,  must  be  satisfied  by  {x^,  y^)  and  [x^.  y^),  must  contain  no 
arbitrary  constant  (Why?),  and  must  not  vanish  identically. 

iy  —  yi)  {^2  —  ^i)=i^  —  ^i)  (y2  —  yi) 
(y—Vz)  {x^  —  x^)={x  —  x.j  (2/1  —  2/2) 


52  Analytic  Geometry. 

meet  these  conditions,  and  either  of  them  may  therefore  be  taken 
as  the  desired  equation.  The  same  result  may  be  obtained  by 
substituting  in 

the  value  of  m  deduced  in  paragrai)h  23.  These  equations  are 
more  frequently  written  in  one  or  the  other  of  the  two  forms 

//  —  l/i  =  ^' ^K^—  ^i) . 


—  l/l  = 

—  //l  - 

_x- 

-  X, 

IL 

Vi  —  Vx       ^2  —  ^x 

Still  another  form  of  this  equation  is  frequently  met  in  mathe- 
matical literature.  It  may  be  deduced  as  follows:  If  the  i)oint 
{x,  y)  is  to  trace  a  straight  line  it  must  satisfy  an  equation  of  the 
form 

ax  -\-l)y  -{-  c^^, 

and  if  the  line  is  to  pass  through  the  points  [x-^,  1/1)  and  {x^,  7/0) 
we  must  have 

ax^  -\-'by^-\-  c==() 
and  aa?2  +  &2/2  +  c  =  0 

The  co-existence  of  these  three  equations  is  the  necessary  and  sufli- 
cient  condition  that  the  point  {x,  y)  may  trace  a  straight  lino^ 
through  the  two  points  {x^,  y^)  and  {x^,  y^).  But  the  necessary 
and  sufficient  condition  for  the  co-existence  of  these  three  equa- 
tions is 

X  y  1 

x^        1/1         1   =0 

a?2         2/2         1 

which  is  therefore  the  equation  of  the  line. 

39-  Consider  a  line  through   (x^,  y^)   making 

THESTRAiGHi  LINE  the  angles  a  and  ^  with  the  axes,  and  let 
GIVEN  BY  TWO  [x,  y)  be  a  variable  point  on  the  line.     The 

EQUATIONS  IN  necessary    and    sufficient    conditions    which 

THREE  VARIABLES,  must  be  Satisfied  in  order  that  the  variable 
point  may  be  on  the  line  are 

X  —  a?i  =  r  cos  a 
and  y  —  yi  =  r  cos  13 


Analytic  Geometry. 


53 


where  r  is  the  variable  distance  from(a?^  y)  to  [x^,  i/J.  These  two 
equations  may  be  written  in  the  form 

cos  a         cos  P 
or  a?  =  ^1  +  r  cos  a 

y  =  yi+r  cos  /3 

These  two  equations  are  used  in  cases  where  the  student  is  inter- 
ested in  the  distance  from  the  tracing  point  to  some  fixed  point 
on  the  line,  r  can  be  readily  eliminated  between  the  two  equations 
and  any  one  of  the  forms  of  the  equation  previously  discussed  at 
once  deduced.  Hereafter  we  shall  refer  to  this  form  as  the  para- 
metric form  of  the  equations  of  the  straight  line. 

40.  The  problem  of  finding  the  distance  from 

DISTANCE  FROM  A    ^  given  point  {x^,  2/i)  to  a  given  line 

POINT  TO  A  LINE. 

Ax-{-By+C  =  0 

might  now  be  solved  by  writing  the  equation  of  a  line  through 
(^i?^i)  perpendicular  to 

Ax-\-By  +  C=0, 

finding  the  point  of  intersection  of  these  two  lines,  and  determin- 
ing the  distance  from  this  point  of  intersection  to  the  point 
1^1?  ^i)-  But  a  geometric  solution  is  simpler  and  leads  to  an 
important  algebraic  result. 

Let  P  be  any  point  (x^,  y^),  and  AB  any  line.  Let  p'  denote  the 
perpendicular  distance  from  the  point  P  to  the  line  AB,  and  p  the 
perpendicular  distance  from  the  origin  to  the  same  line.  Let  a 
and  'P  denote  the  angles  which  the  perpendicular  from  the  origin 


Fig.  15. 


Fig.  16. 


on  the  line  AB  makes  with  the  axes.    Draw  Pa  perpendicular  to 
OX,  Ph  and  Od  perpendicular  to  AB,  and  ac  parallel  to  AB. 


54  Analytic  Geometry. 

Then  Oc -\-hP —  Od  =  Fe 

or  Oc-\-hP—Od  =  —  PG 

i.  e.,  x^  cos  a  +1/1  sin  a  — p  =  p   ov  — p 

according  as  the  point  F  is  on  the  opposite  or  same  side  of  the  line 
AB  as  the  origin.  This  formula  gives  the  distance  from  the  point 
to  the  line  in  terms  of  the  co-ordinates  of  the  point  and  the 
constants  p  and   a   which  determine  the  position  of  the  line. 

41.  The    necessary    and    sufficient    condition 

NORMAL  FORM  OF  that  any  point  {co,  y)  be  on  the  line,  (i.  e. 

THE  EQUATION  OF  the  equation  of  the  line)  is  that  p   for  thai 

A  STRAIGHT  LINE,  point  shall  be  zero. 

X  cos  o.  -\-y  sin  a  — p  =  0 

is  therefore  another  form  of  the  equation  of  the  straight  line.  It 
is  called  the  normal  form  of  the  equation. 

In  order  to  reduce  the  general  equation  to  the  normal  form  we 
multiply  it  by  an  undetermined  constant  m.    Then  if 

mAx  +  mBy  +  mC^='  0 
is  identical  with 

X  cos  a  -|-  y  sin  a  —  p  =  0 
we  must  have 

mA  =  cos  a 
mB  =  sin  a 
m  C  =  —  p 
Therefore  m^A^  +  m^i?^  =  1 

and  m  ^  — =:zzzz=r  • 

l/A'  -f  B' 

Substituting  this  value  we  have  for  the  normal  form  of  the  equa- 
tion 

Ax      _^         By       _^  C       ^  ^ 


]A'-\-B'      lA'  +  B'      j/A'-hB' 

The  results  of  the  last  two  articles  may  evidently  be  stated  in 
the  following  condensed  form.  The  distance  of  any  point  from  a 
given  straight  line  is  the  value  obtained  by  substituting  the  co- 
ordinates of  the  point  in  the  left  hand  member  of  the  normal 
form  of  the  eiiuation  of  the  line. 


Analytic  Geometry. 


55 


PROBLEMS. 


1.  Write  the  equations  of  each  of  the  following  lines  in  each 
of  the  forms  already  developed. 


Through  the  point 

Angle  with  X  axis. 

(1,  3) 

IT 

(1,  i) 

3  radians 

(-2,-4)     ■ 

65  degrees 

{a,  b) 

c  degrees 

Through  the  Points 

(1,  4)      and     (2, 



3) 

(1,-6)                (a, 

/?) 

(4,  c)                 (0, 

2) 

(4,  5)                 (4, 

3) 

(4,  2)                 (- 

-a, 

2) 

Through  the  origin  parallel  to  the  first  of  the  list. 
Through  (1,  1)  perpendicular  to  the  first  of  the  list. 

2.  Find  the  distance  from  the  origin  to  each  of  the  lines  in  the 
first  group  above. 

3.  Find  the  distance  from  the  i)oint  (1,  1)  to  each  of  the  lines 
of  the  first  group  above.  In  which  cases  are  the  origin  and 
(1,  1)  on  opposite  sides  of  the  line? 

4.  Find  the  area  of  the  triangle  whose  vertices  are  (0,  0), 
(1,2),  (3,1). 

5.  Find  the  area  of  the  triangle  whose  vertices  are  (1,  — 1), 
(3,2),  (-3,-2). 

6.  If  A  is  the  area  of  a  triangle  and  {x^,  y^),  (x^,  y^),  (x^,  y^\ 
are  its  vertices,  show  that 

2A  =  y^{x^—x^)+y^(x^~x^)  +  y^(x^—x^) 

7.  In  determinant  notation  this  becomes 


2A  = 


Deduce  this  form  directly  by  use  of  the  determinant  form  of  the 
equation  of  a  straight  line. 

8.  To  say  that  three  points  lie  on  a  straight  line  is  evidently 
equivalent  to  saying  that  the  area  of  the  triangle  formed  by  them 
is  zero.  Show  that  the  necessary  and  sufficient  condition  for 
collinearity  of  three  points  {x^,  y^),  {Xo,  y^),  i^s^  Vz)  is 

1 


2/i 

1 

2/2 

1 

Vz 

1 

a?3        2/3         1 


0. 


56  Analytic  Geometry. 

9.  Deduce  the  same  conclusion  from  the  fact  that  the  necessai'y 
and  suflicient  condition  that  the  three  points  may  lie  on  the  line 

Ax-{-By  -{-C  =  {) 

is  that  the  co-ordinates  of  each  point  shall  satisfy  the  equation  of 
the  line. 


42. 
INTERSECTIONS  OF 
LINES. 


PROBLEMS. 

Find    the   intersections   of   the    following 
lines : 


1.  3x  —  2y-\-2  =  Q  and  w-\-y  =  2 

2.  w  —  ^y +  1  =  0  and  x  — Ay  — 1  =  0 

3.  w  =  2y  —  7  and  y^lx  —  2 

4.  Find  the  vertices  of  the  triangle  formed  by  the  lines 

2a?  — 3^  =  4 

Sa-  — 4?/+2  =  0 

5.  Show  that  the  necessary  and  sufficient  condition  which  must 
be  satisfied  in  order  that  the  three  lines 

A,x+B,y  +  G,  =  0 
A^x  +B^y  +  C^  =  0 
A,x+B,y+C,  =  0 

may  meet  in  one  point  (i,  e.,  be  concurrent)  is 

^0 


A,        B,         C, 
Ao         B._         C. 


In  general  any  two  lines  intersect  in  one  point.  The  only  case 
in  which  an  apparent  exception  is  to  be  noted  is  when  the  lines 
are  parallel.  It  is  possible  to  include  this  apparent  exception, 
however,  under  the  general  case.  If  the  student  will  refer  to  his 
discussion  of  Problem  12,  Paragraph  36,  and  take  into  consider- 
ation the  geometric  significance  of  m^  and  Wo  he  will  see  that  the 
result  of  that  discussion  may  be  stated  thus.  As  two  lines  tend 
to  parallelism  one  or  both  of  the  co-ordinates  of  the  point  of 
intersection  must  increase  beyond  all 'limit,  i.  e.,  the  point  of  inter- 
section removes  indefinitely  from  the  origin,  Mathematicians  are 
accustomed  to  use  in  place  of  this  statement  the  abbreviated 
phrase,  ''Parallel  lines  intersect  at  infinity''  and  in  this  way  are 
able  to  include  the  special  case  of  parallelism  in  the  general 
statement  that  any  tw^o  lines  intersect  in  a  single  point. 


Analytic  Geometry.  57 

42-  Let  the  straight  lines  represented  by  the 

FAMILIES  OF  LINES,  two  equations 

and  A^x  +  B^y  +  O2  =  0 

Intersect  in  the  point  {x^,  yj.  Then  if  /x  and  v  are  any  two 
constants, 

f.(A,w  +  B,y  +C,)+  V  {A,w  +  B,y  +  C,)  =  0 

is  the  equation  of  a  straight  line  (Why?)  passing  through  the 
point  (a?i,  7/1),  since  the  substitution  of  x^  and  1/1  for  a?  and  y 
causes  each  of  the  parentheses  to  vanish  and  therefore  satisfies 
the  equation.    If  we  put 

A,x-{-B,y-i-C,  =  8, 
A,x-\-B,y+C,=  S, 

we  may  write  the  third  line  in  the  abridged  form 

In  general  if  7?^  :=  0  and  R.  -~-0  are  the  equations  of  any  two  loci. 

is  the  equation  of  a  third  locus  passing  through  all  the  inter- 
sections of  the  first  and  second. 
The  equation 

/Sf,  +  \S^  =  0 

contains  one  arbitrary  constant  as  it  properly  should,  since  the 
line  represented  by  it  has  been  subjected  to  the  single  condition 
of  passing  through  one  point  and  therefore  has  one  degree  of 
freedom  remaining.  This  constant  may  be  determined  so  that  the 
line  passes  through  any  other  point  in  the  plane,  and  therefore 
it  is  evident  that  the  equation  by  proper  choice  of  X  may  be 
made  to  represent  any  line  in  the  plane  through  the  intersection 
of  ^S\  =  0  and  aS^  =  0.  The  aggregate  of  all  such  lines  is  spoken 
of  as  the  family  of  lines  through  the  intersection  of  >S'i=0  and 
^*2  =  ^  and  the  equation 

is  called  the  equation  of  the  family.     The  arbitrary  constant  is 


58  Analytic  Geometry. 

called  a  parameter,  a  name  given  to  any  constant  entering  into 
an  equation  and  taking  in  the  course  of  the  discussion  a  succession 
of  arbitrary  values.    Thus 

y  =  a 

y  —  y^  =  n{x  —  x^) 
are  respectively  the  equations  of  the  families  of  lines  which  pass 
through  the  origin,  are  parallel  to  the  X  axis,  and  pass  through 
the  point  (a?i,  y^).  The  respective  parameters  are  m,  a,  and  n. 
If  it  is  desired  to  determine  a  particular  member  of  a  family 
an  additional  condition  must  be  given.  When  the  equation  of 
the  family  is  subjected  to  this  condition  a  value  or  set  of  values 
of  the  parameter  will  be  determined  which  will  determine  the 
member  or  members  of  the  family  satisfying  the  condition.  For 
example,  suppose  we  wish  to  determine  that  member  of  the  family 

A,x-{-B,y  +  C^+X{A,x^B,y-^C^)=i) 
which  passes  through  the  point  (3,  2).    We  have  at  once 

3Ai  +  2B^  +  C\  +  X(3A2  +  ^^2  +  ^2)-=  0 
whence  the  value  of  X  is  at  once  determined.     Substituting  this 
value  of  A  in  the  equation  of  the  family  we  have  the  particular 
member  of  the  family  which  we  desire. 

PROBLEMS. 

1.  Write  the  equation  of  the  family  of  lines  through  the  inter- 
section of 

2a?  —  3i/  +  1  =  0 
and  5a?  — i/-f2==0 

and  find  that  member  of  the  family  which 

(a)  passes  through  the  point  (1,  5) 

(b)  is  perpendicular  to  4a?  —  2i/  =  0 

(c)  has  an  intercept  of  4  on  the  X  axis 

(d)  makes  an  angle  of  30  degrees  with  the  X  axis. 

(e)  is  parallel  to  the  bisector  of  the  1st  and  3rd  quadrants  at 
the  origin. 

2.  Write  the  family  of  lines  through  the  point  (1,  2). 

3.  Write  the  family  of  lines  parallel  to  the  line 

3a?  +  2i/  +  4  =  0. 

44.  If  the  equation  of  a  line  be  written  in  the 

THE  LINE  AT  Intercept  form 

INFINITY.  '!Lj^2L=^\ 

a        b 
and  the  line  be  removed  farther  and  farther  from  the  origin,  a 


Analytic  Geometry.  59 

and  h  indefinitely  increase  and  the  coefficients  of  x  and  y  tend  to 
zero.  In  other  words,  as  a  line  removes  indefinitely  from  the  origin 
its  equation  tends  to  the  form 

0.V  +  Of/  =  1 

or,  since  the  equation  may  be  multiplied  by  any  constant  what- 
ever, to  the  form 

Ox -}-()!/ =C. 

Moreover  this  form  does  not  depend  on  the  original  position  of 
the  line  or  on  the  manner  in  which  it  is  removed.  In  other  words 
the  equations  of  all  lines  tend  to  a  single  form  as  the  lines  are 
indefinitely  removed  from  the  origin,  a  statement  which  is  equiv- 
alent to  the  assertion  that  all  points  at  infinity  satisfy  the  single 
first  degree  equation 

0^  +  0//  =  C. 

But  since  in  analytic  geometry  we  deal  with  loci  only  'through 
their  equations,  mathematicians  are  accustomed  to  express  all 
this  by  the  abbreviated  phrase,  ''All  points'  at  infinity  lie  on  a 
single  straight  line  whose  equation  is 

0,^  +  0//=  C' 

For  any  point  in  the  finite  part  of  the  plane  the  left  hand  member 

of  the  above  equation  is  of  course  zero,  and  since  we  are  in 

almost  all  of  our  investigations  concerned  only  with  the  finite 

part  of  the  plane,  it  is  customary  in  all  such  investigations  to 

abbreviate  still  more  the  above  form  and  write  the  equation  of  the 

line  at  infinity  as 

(7  =  0. 

45.  The  subject  of  parallel   lines   affords  an 

PARALLEL  LINES.       interesting  application   of  this  idea  of  the 
line  at  infinity.     Let  the  line  GH  be  drawn 
through  the  intersection  of  AB  and     a  ^2) 

CD.  Then  if  the  equations  of  ^^and 
CD  are  respectively 

S^  =  0  and  8^  =  0 
the  equation  of  GH  is 

Let  CD  be  removed  indefinitely,  its  equation  will  tend  to  (7  =  0. 


60  Analytic  Geometry. 

OH  will  tend  to  jjarallelism  with  AB,  and  its  equation  will  tend 
to  the  form 

which  gives  our  former  theorem  that  the  equation  of  a  line  parallel 
to  a  given  line  differs  from  the  equation  of  the  given  line  only  in 
the  constant  term.  A  line  parallel  to  a  given  line  is,  from  this 
new  point  of  view,  a  line  through  the  intersection  of  the  given  line 
with  the  line  at  infinity. 


CHAPTER  XI. 

THE  CIRCLE,  A  SPECIAL  CASE  OF  THE  EQUATION  OF 
THE   SECOND  DEGREE. 

46.  The  most  general  form  of  the  equation  of 

THE  GENERAL  ^j^g  second  degree  is 

EQUATION  OF  THE 

CI  RCLE.  «^'  +  W  +  2/ia?i/  +  2^a?  +  2/1/  +  c  =  0 

and  the  corresponding  curves,  from  certain  relations  which  they 
bear  to  the  cone,  are  called  conic  sections  or  simply  conies. 

Before  taking  up  the  study  of  the  general  conic  we  shall  con- 
sider the  special  case  of  the  circle.  Its  equation  has  been  already 
found  (Problem  3,  Paragraph  24),  in  the  form 

where  (a,!:))  is  the  center  and  r  the  radius.  Multiplying,  trans- 
posing, and  replacing  the  constant  term  by  c,  w^e  reduce  this 
equation  to  the  form 

a?2  _|_  2/'  —  2aa?  —  2hy  +  G=-^ 
or  multiplying  by  an  arbitrary  constant 

Ax""  +  Ay-"  +  2Gx  +  2Fy  +(7  =  0. 

We  therefore  state  the  general  theorem  that  every  circle  is  repre- 
sented by  a  second  degree  equation,  without  a  term  in  xy,  and 
with  the  same  coefficient  for  the  terms  in  x^  and  y"^.  The  student 
may  deduce  the  converse  theorem  that  every  second  degree  equa- 
tion without  a  term  in  xy  and  with  the  same  coefficients  for  the 
terms  in  x^  and  y^  represents  a  circle  by  showing  that  every  such 
equation  may  be  reduced  to  the  form 

(x  —  aY-^{y  —  'by  =  r\ 

PROBLEMS. 
1.  Compare  the  two  forms 

Ax'-  +  Ay^  +  2Gx  +  2Fy  +  0  =  0 

and  (x  —  a)'^ -\- {y  —  fej^^r^ 

and  hence  show  that  the  center  is  the  point    ( , -)  and 


A'       A 


1 


the  radius  is  —  V(>'  +  ^'  —  ^O. 
A 


62  Analytic  Geometry. 

2.  Find  the  centers  and  radii  of  the  following  circles : 

x^  -\-  qf  —  4:X  -\-  5y  —  12  =  0 
Sx''  +  Sy''  —  2y=0 

Ax'  +  4^2  _i2x-\-2y  +  4=--0 
3a?2  +  32/2  — 4a?  +  2  =  0 

^^  -{-  y^  -\~  p^  -{-  QV — s  =  o 

3.  Fiifd  the  equations  of  the  circles  with  the  following  centers 
and  radii. 

Centers  Radii 

(3,  4)  2 

(4,  1)  I 

(3, 0)  5 

(-2,-1)  k 

(0,-4)  sin - 


4.  Write  equations  which  by  proper  choice  of  the  constants  in- 
volved will  represent  any  circle  of  radius  2  whose  center  is  on  the 
X  axis,  on  the  line  y^=a,ovL  the  line  y  =  ^x,  on  the  line  t/  =  3^  +  2, 
on  the  circle  x^  -f-  1/2  =  4. 

5.  Write  the  equation  of  the  circle  centered  on  the  curve  y^  =  3j? 
and  passing  through  the  points  (2,  3)  and  (4,  4). 

6.  Apply  to  the  circle 

X-  -]-y^  =zr 

a  transformation  of  co-ordinates  which  will  make  both  the  new 
axes  tangent  to  the  circle.    What  is  the  new  form  of  the  equation  ? 

7.  What  is  the  form  of  the  equation  when  the  new  axes  are  a 
diameter  and  the  tangent  at  its  extremity  ? 

8.  Write  the  equation  of  a  circle  centered  at  (3,  1)  and  tangent 
to  the  line 

^x  —  2y  +  4.  =  0 

(See  paragraph  41.) 

9.  Show  by  .a  geometric  construction  that  the  condition 

{x  —  a)^+iy  —  h)^  =  r^ 

is  satisfied  by  the  co-ordinates  of  all  points  on  the  circle  of  radius 
r  centered  at  ia,h)  and  by  no  others. 

47.  We  might  find  the  intersections  of  two 

INTERSECTIONS  OF    circles  by  a  direct  solution  of  tUe  two  equa- 
ciRCLES.  tions  for  x  and  y,  but  this  would  introduce 

an  unpleasant  amount  of  algebraic  work.    Subtracting 

x'-{-if+2a,x  +  2h,y+c,  =  0 
from  x^-{-y^ +  2a2X  +  21)r,y +  0^  =  0 

we  have        2x{ao  —  a^)-\-2y{l)2  —  ?>i)  +  02  —  c^  =  0, 


Analytic  Geometry.  63 

a  new  locus  passing  through  the  intersections  of  the  two  circles. 
(See  Paragraph  43.)  This  locus  is  a  straight  line  and  must 
therefore  be  the  common  chord  of  the  two  circles.  Our  problem 
is  now  reduced  to  the  simpler  one,  already  solved,  of  finding  the 
intersections  of  this  line  and  either  of  the  circles.* 

When  one  of  the  circles  lies  wholly  within  or  without  the  other 
the  intersections  are  of  course  imaginary.  It  will  probably  sur- 
prise the  student  to  find  that  the  common  chord  is,  however, 
always  real.  The  explanation  of  the  fact  will  be  evident  if  he  will 
find  the  equation  of  the  line  determined  by  a  pair  of  conjugate 
imaginary  points,  e.  g.  {a  -{-ih^  c -\-  id)  and  {a  —  tb^  c — id). 

PROBLEMS. 

Find  the  intersections  of  the  following  pairs  of  circles : 

1.  x^  -\-  y^  +  2x  —  3y  +  2  =  0 
(a?  —  3)^+(7/  — 4)2  =  16 

2.  {x  —  l)-'+{y— 2)^  =  4: 
x^  -\-  y^  ^4: 

3.  (^—l)2  +  (2/— 2)^  =  4 
(^  +  3)^+(t/  +  5)2  =  l 

4.  (a?— 6)2+(i/  — 4)2=4 
{x-2)^+{y  — 1)^  =  9 

48.  While  the  definition  sometimes  given  in 

TANGENTS  AND  elementary  geometry  of  a  tangent  line  to  a 

NORMALS.  circle  as  a  perpendicular  to  a  radius  at  its 

extremity  is  entirely  correct  and  might  be  used  as  a  basis 
for  our  discussion  of  the  tangent,  it  is  not  a  definition 
which  admits  of  extension  to  other  curves  which  we  shall 
study.  The  tangent  line  to  any  curve  may  evidently  be 
regarded  as  the  limiting  position  of  a  secant  line  as  two  of 
the  points  of  intersection  of  the  curve  and  the  secant  tend 
to  coincidence.  Let  a  secant  line  meet  a  curve  in  two  points  P 
and  Q.  and  let  the  point  P  tend  to  coincide  with  Q.    The  limiting 


*Two  circles  have  of  course  four  points  of  intersection  (paragraph  35), 
but  two  of  them  are  always  imaginary  points  at  infinity.  The  subtraction 
above  gave  terms  Ox^  and  O1/2.  To  drop  these  as  we  did  was  to  assume  that 
X  and  y  were  to  remain  finite,  and  the  resulting  equation  therefore  is  satis- 
fied by  the  finite  intersection  of  the  circles,  but  not  by  the  infinite  ones. 


64  Analytic  Geometry. 

position  of  the  secant  line  as  P  tends  to  Q  is  the  tangent  at  Q. 
This  is  sometimes  expressed  by  saying  that  the  tangent  to  a  curve 
meets  it  in  two  coincident  points. 

.  The  normal  to  a  curve  at  any  point  is  the  perpendicular  to  the 
tangent  at  that  point. 

Among  the  questions  that  arise  concerning  tangents  are  two 
that  decidedly  outrank  the  others  in  importance. 

1.  Given  the  equations  of  a  line  and  a  circle,  how  shall  we 
determine  whether  the  line  is  tangent  to  the  circle,  or  in  other 
words,  what  is  the  condition  which  the  coefficients  of  the  line 
must  satisfy  in  order  that  it  may  be  a  tangent  line  to  the  circle? 

2.  Given  the  co-ordinates  of  a  point  and  the  equation  of  a 
circle,  what  is  the  equation  of  a  line  through  the  point  tangent 
to  the  circle  ? 

49.  Any  satisfactory  definition  of  the  tangent 

CONDITION  OF  ^ju  ^f  course  lead  to  the  condition  of  tan- 

^"^    ^"^      *  gency  if  properly  considered.    For  example, 

the  fact  that  the  tangent  is  perpendicular  to  the  radius  at  its 
extremity  is  equivalent  to  the  statement  that  the  perpendicular 
distance  from  the  point  («,?>)  to  the  line 

y  =  mx  +  h 

must  equal  r  if  the  line  is  tangent  to  the  circle  of  radius  r  centered 
at  {a,!)).    Applying  this  test  we  have  as  the  condition  of  tangency 

ma  —  &  -f-  h 

:  =  r 


or  /^=zbrVl+w^^ — ma-\-}) 

This  method  of  deriving  the  condition  of  tangency  is  unfortu- 
nately applicable  only  to  the  circle,  since  the  definition  of  tangency 
on  which  it  is  based  does  not  hold  for  other  curves.  The  definition 
of  the  tangent  as  a  line  meeting  the  curve  in  two  coincident  points 
holds  however  for  all  curves.  In  deducing  the  condition  of  tan- 
gency from  this  definition  we  find  the  intersections  of  the  line  and 
the  curve.  The  co-ordinates  of  these  intersections  are  given  by 
ordinary  algebraic  equations  in  one  variable  and  the  necessary 
and  sufficient  condition  for  tangency  is  that  these  equations  shall 
have  equal  roots.    Thus 

y  =  mx  -\-  h 
meets  x^  -\-  y^=^  r- 


Analytic  Geometry.  65 

in  two  points  whose  a?  co-ordinates  are  given  by  the  quadratic 
equation  x^  +  ^^^^^  +  2ma?/i  +  ^^^  —  r^  =  0. 

The  necessary  and  sufficient  condition  for  coincidence  of  the 
points  of  intersection,  and  therefore  for  tangency  of  the  line,  is 
that  this  quadratic  in  ^r  be  a  perfect  square,  i.  e.,  that 

or  /^^  it  r\/l -|- m^ 

as  before. 

50.  Given  a  point  {x-^,  y^)  and  a  circle 

EQUATION  OF  THE  0,0    o 

TANGENTTHROUGH  '  ^      ' 

A  GIVEN  POINT.  to  determine  the  equation  of  the  tangent  to 

the  circle  through  the  given  point  we  first 
write  the  general  equation  of  a  line  through  the  point 

y  —  y^  =  m{x  —  x^) 
or  y^  mx  +2/1  —  mx^. 

If  this  line  is  tangent  we  must  have  from  the  last  paragraph 


7/1  —  mx^  =  dz  r\/l  +  nv 


x,y^  ±r\/x^^  -\-y^^  —  r^ 

Inserting  these  two  values  of  m  in  the  equation  of  the  line  we 
have  the  equations  of  the  two  tangents  from  (a?i,  y^)  to  the  circle. 
If  however  (x-^^,  yj  is  on  the  circle,  we  have 

x,^  +  y,'-r^=0,  -        . 

and  therefore  m= ^ 

Substituting  this  value  of  m  in  the  equation  of  the  line  we  have 
for  the  equation  of  the  tangent  at  a  point  {x^,  yj  on  the  circle 

x^  -{-  y^  =  r^ 

the  form  y  —  yi  =  —  —  (^  —  ^1) 

i.  e.,  yyi  +  («^i  =  r^ 

This  method  of  finding  the  equation  of  the  tangent  is  theoretic- 
ally general,  but  in  the  case  of  more  complex  equations  we  en- 
counter serious  algebraic  difficulties.  For  the  development  of  th«3 
equation  of  the  tangent  at  a  point  on  a  curve  a  second  method, 
based  upon  the  definition  of  the  tangent  as  the  limiting  position 


66 


Analytic  Geometry. 


of  the  secant,  is  worth  our  investigation.    Consider  then  a  point 
Pf  (a?i,  1/i),  on  the  circle,  and  give  to  x^  and  y^  such  increments 
Aa?i  and  A?/i,  that  the  new  point 
Q,  (iPi+  Ao^i,  2/1+  A2/1),  so  ob- 
tained shall  also  be  on  the  circle. 
Then  the  line 

y—y^  =  ^{^  —  ^^) 

is  the  secant  line  through  the  two 
points  (£i?i,  1/1)  and  (a?i  +  A^i, 
1/1  -|-  A2/i)-    If  we  can  determine 

the  limiting  value  of  the  ratio  — ^ 

Aa?i 

as    Aa?i    and     consequently    A//i 

tend  to  zero  we  shall  have  the  ^^^-  •^^• 

slope  and  therefore  the  equation 

of  the  tangent  at  [x^,  y^).    Since  both  points  are  on  the  circle  we 

have 

and        a?i'  +  2x,  /\x,  +  A^'  +  y'  +  2y^  Ay,  +  A^i'  =  r' 
Subtracting  the  first  of  these  equations  from  the  second  we  have 


2.z?iA^i  +  Axi  +  2^iA//i  +  A//i"  =  0 

whence  Ay, 2.^1  +  A^i 

Ax,  ~~      2//i  +  Ay, ' 


But  the  limit  of  this  fraction  as  A-^i  tends  to  zero  is  —  -^  which 

is  therefore  the  limiting  value  of  the  slope  of  the  secant  line 
through  the  two  points  as  the  second  point  tends  to  coincidence 
with  the  first,  i.  e.,  the  slope  of  the  tangent  at  {x,,  y,).  The  equa- 
tion of  the  tangent  is  therefore 

y  —  y,=  —  ^  {x  —  x,) 

l/i 
reducing  as  before  to 

yy^-{-xx,  =  r^ 

This  method  is  that  employed  in  the  differential  calculus,  and 
by  the  aid  of  the  processes  elaborated  in  the  discussion  of  that 
subject  is  applicable  to  the  more  complex  forms  which  present 
too  much  algebraic  difficulty  for  our  former  method. 


Analytic  Geometry. 


67 


51.  The  distance  from  the  point  of  tangency 

SUB-TANGENT  AND    to  the  point  where  the  tangent  intersects  the 
SUB-NORMAL.  X  axis  is  called  the  length  of  the  tangent, 

and  the  projection  of  this  portion  of  the  tangent  line  on 
the  X  axis  is  called  the  sub-tangent.  The  distance  from 
the  point  of  tangency  to  the  point  where  the  normal  in- 
tersects the  X  axis  is  called  the  length  of  the  normal, 
and  the  projection  of  this  portion  of  the  normal  line  on 
the  X  axis  is  called  the  sub-normal.  Thus  if  the  center  of  the 
circle  be  at  the  origin,  FT,  PO,  OQ  and  QT  are  respectively  the 
lengths  of  the  tangent,  normal,  sub-normal  and  sub-tangent  at  P. 
Let  the  radius  of  the  circle  be  r;  the  co-ordinates  of  P  be  a?i,  i/i ; 
and  of  T  be  x^,  0.  Then  in  this  special 
case  it  is  evident  that  the  lengths  of 
the  normal  and  sub-normal  are  r  and  Y 

Xj^.     To  find  the  lengths  of  the  tan 
gent  and  sub-tangent  we  write  the 
equation  of  the  tangent  at  P  and  find 
its  intersection  with  the  X  axis.    The 
length  of  the  sub-tangent  is  then 

2 2  2 

x^  —  w^  =  ^- ^  =  ^, 

X  Xi 

and  the  length  of  the  tangent  is 


V{^2  —  ^i)^-i-yi 


■4^^' 


1  '  ") 

X, 


Fig.  19, 


results  which  might  have  been  directly  obtained  by  the  aid  of  trigo- 
nometry. 

PROBLEMS. 


1.  What  is  the  condition  of  tangency  to  the  circle  when  the 
equation  of  the  line  is  given  in  the  form 

2.  Find  the  equations  of  the  tangents  to 

x^  -\-y^  ^4: 
through  (1,3),  (5,  6),  (1,-1). 


68  Analytic  Geometry. 

3.  Find  the  tangents  to 

through  the  point  (5,  3). 

4.  Show  from  the  quadratic  for  determining  m  that  the  tangents 
through  P  are  real  or  imaginary  according  as  P  is  outside  or 
inside  the  circle. 

5.  Show  that  the  equation  of  the  tangent  to 

x\-\-  if  -\-2ax  +  2'hij  +  c  =  ^ 
at  the  point  {x^,  2/i)  is 

y-yr=~  "^^^{x-x,). 

This  form  reduces  to 

a^^i  +  2/2/i  +  «^  +  ^^  =  ^i'  +  ^i'  +  «^i  +  ^Vi 
Adding         axy  -j-  hy^  -\- c  to  both  sides,  the  second  member  van- 
ishes (why?)  and  we  have  a  frequently  used  form 

6.  Show  that  the  equation  of  the  normal  to  the  circle 

x"^  -\-  y'^  =  r^ 
at  the  point  (x^,  :?/i)  is 

xy^  —  yx^  =  0. 

Note  that  the  normal  to  the  circle  always  passes  through  the 
center. 

52.  Given  any  tw^o  tangents  to  a  circle,  the 

POLES  AND  POLARS  chord  joining  their  points  of  tangency  is 
DEFINED.  called  the  chord  of  contact.     We  have  just 

seen  that  any  point  determines  two  tangents  to  a  circle 
and  hence  it  determines  a  chord  of  contact.  Similarly 
any  chord  of  a  circle  determines  the  two  tangents  at  its 
points  of  intersection  with  the  circle,  and  hence  it  de- 
termines a  point,  the  intersection  of  the  two  tangents.  In  other 
words  there  exists  a  one  to  one  correspondence  between  the  chords 
of  any  circle  and  the  points  of  the  plane,  so  that  to  each  point  there 
corresponds  a  single  chord  and  conversely.  The  chord  is  called 
the  polar  line  or  merely  the  polar  of  the  point  with  respect  to  the 
circle,  and  the  point  is  called  the  pole  of  the  line  with  respect  to  the 
circle. 


Analytic  Geometry. 


69 


53. 


EQUATION  OF  THE 
t  OLAR. 


The   equation   of  the   polar   of   the   point 
(^1,  7/J  with  respect  to  the  circle 

x~  +  y~  =  r^ 

might  be  derived  directly  by  finding  the  points  of  contact  of  the 
tangents  from  {x^,  i/J  and  writing  the  equation  of  the  line  through 
the.^e  two  points ;  but  the  algebraic  work  is  somewhat  complicated 
and  we  will  accordingly  make  use  of  a  method  similar  to  that  of 
paragraph  38. 

Let  F  be  any  point  {x^,  y-^)  and 
let  the  two  tangents  from  P  to  the 
circle 

x"^  +  y"^  ='■  "^^ 

touch  the  circle  at  the  points  A  and 
^j  (^2?  2/2)  ^^d  (a?3,  1/3).  Since  PA 
is  tangent  at  (x^,  i/2)  its  equation  is 

xx^  +  2/2/2  =  r^ 
Similarly  the  equation  of  PB  is 

^^3 +  2/2/3  =^'- 
But    each     of    these     lines    passes 
through    (a?i,   1/1 )    and  therefore  we 
have 

«^1^2  +  2/12/2=^'' 

^1^3  +  2/12/3  =  r\ 
The  problem  before  us  is  to  find  an  equation  of  the  first  degree 
in  X  and  y  which  is  satisfied  when  x  and  y  are  replaced  either  by 
Xr,  and  y.y  or  by  x^  and  y^.    An  inspection  of  the  pair  of  equations 
last  written  shows  that 

^^1  +  2/2/1  =  ^^ 
is  such  an  equation.     It  is  therefore  the  equation  of  the  polar 
of  the  point  (^1,  2/1)  with  respect  to  the  circle 

x~  -\-  y"^  =  r-. 

When  the  point  (x^^,  y^)  is  on  the  circle  the  equation  of  the  polar 
bf^comes  the  equation  of  the  tangent  at  (a?j,  y^).  In  other  words 
the  tangent  is  only  a  special  case  of  the  polar,  being  the  polar  of 
the  point  of  tangency.  By  allowing  the  point  P  in  the  figure  to 
approach  the  circle  the  student  can  convince  himself  that  the 
tnngent  is  the  limiting  position  of  the  polar  as  the  pole  approaches 
the  circle. 


Fig.  20. 


70 


Analytic  Geometry. 


The  discussion  given  above  in  no  way  depends  on  the  location 
of  the  point  P  outside  the  circle.  If  P  is  inside  the  circle  the  two 
tangents  are  imaginary  and  the  points  of  contact  also  imaginary ; 
but  if  the  pole  P  and  the  circle  are  real  the  points  of  contact  are 
conjugate  imaginary  points  and  the  polar  is  real,  a  fact  which 
is  also  evident  from  the  equation. 

54.  To  find  the  pole  of  a  given  line  with  respect 

CO-ORDINATES  OF     to  the  circle 
THE  POLE.  x^-\-y'^  =  r'^ 

we  may  also  use  a  method  shorter  than  the  direct  one  of  finding 
the  intersection  of  the  two  tangents  having  the  given  line  as  chord 
of  contact.    Let  the  given  line  be 

ax  -{-Ijy  -\-c  =  ^ 
and  assume  the  co-ordinates  of  its  pole  to  be  {x^,  i/J.    Then  the 
equation  of  the  line  must  be 

Since  these  tw^o  equations  represent  the  same  line  their  co- 
efficients must  be  proportional  (why  not  equal?)  i.  e., 


whence 


—  Ml  — 

—  r 

h 

c 

-  a/ 

Ih 

hr' 


55. 
POLAR  AS  LOCUS 
OF  HARMONIC 
CONJUGATES. 


C  G 

Consider  any  point  P  and  draw  through  P 
a  line  meeting  the  circle  in  the  points  Q  and 
R.  Let  H  be  the  harmonic  conjugate  of  P 
with  respect  to  Q  and  R.  The  locus  of  S  as 
the  line  rotates  through  P  is  the  polar  of  P  with  respect  to  the 
circle. 

The  proof  of  this  theorem  assumes  the  following : 

(1)  Problem  7,  paragraph  9 ; 

(2)  The  roots  of  ex-  -\-hx  -\-  a  =  y) 
are  the  reciprocals  of  the  roots  of 
ax^  +  &  a?  +  c  =^  0 ; 

(3)  The  sum  of  the  roots  of  ax-  -\-  hx 

+  c  =  0  is  — -^. 
a 

Let  the  co-ordinates  of  P  be  (^i,  i/i) 
and  the  equation  of  the  circle  be  fig.  21. 

x^  -\-y^z=r^ 


Analytic  Geometry.  71 

Write  the  equation  of  the  line  througli  P  in  the  form 

x  =  x-^-\-  p  cos  a 

Substitute  these  values  of  x  and  y  in  the  equation  of  the  circle, 
and  the  distances  FQ  and  PR  from  the  point  P  to  the  intersections 
of  the  line  and  the  circle  are  given  by  the  quadratic 

p'  (cos'  a  -j-  cos'  /?)  +  2p  Cxi  cos  a  +  t/^  cos  /S)  +  x^^  -}-  y^^  —  /  =  0 

n  1      ,      1    _      2CT,cosa4-yiCos/3) 

^""^  FQ^PR"  x^-^y^-7^ 

Let  the  co-ordinates  of  ^  be  {x  ,  y')  then 


The  necessary  and  sufficient  condition  that  /Sf  shall  be  the  harmonic 
conjugate  of  P  with  respect  to  Q  and  R,  i.  e.,  the  equation  of  the 
locus  of  B,  is  therefore 

—  2  (x^  cos  g  +  j/i  cos^) 2 

But    cosa=  ^^       "^'^ 


i/(y  — e^g'  +  (v/'-//ir 


cos  /? 


1/  —  ?/l 


1/  {x'  —  xS^  +  {y'  —  yy 
Substituting  these  values  we  have  for  the  equation  of  the  locus 

^'^1  +  yyi  =  ^' 

i.  e.,  xx^  +  yy,  =  r^ 

as  before. 

56.  An  important  problem  here  presents  itself. 

POLAR  AS  LOCUS  Given  a  fixed  point  and  a  circle,  to  find  the 
OF  POLES.  locus  of  the  poles  with  respect  to  the  given 

circle  of  all  lines  through  the  given  point.  This  locus  is  evidently 
a  definite  curve  and  therefore  its  equation  must  be  a  single  re- 
lation between  the  co-ordinates  of  the  variable  point  and  known 
constants,  but  without  arbitrary  parameters.  Let  the  point  be 
(a?i,  i/i)  and  the  circle  be  given  in  the  form 

x^  -\-y~  =  r^. 
The  equation  of  any  line  through  the  point  is 
y—y^  =  m{x  —  x,) 


72  Analytic  Geometry. 

and  the  co-ordinates  of  the  pole  are 


V  =  — 7 — 

—  mxi  +  i/i  —  iiu\  +  f/i 

We  have  here  two  equations  connecting  the  co-ordinates  of  tho 
variable  point  with  known  constants  and  with  the  arbitrary  para- 
meter m.  What  we  desire  is  a  single  relation  free  from  arbitrary 
parameters,  connecting  the  co-ordinates  with  each  other  and  wiin 
known  constants.  We  therefore  eliminate  m  and  find  for  the  equa- 
tion of  the  desired  locus  (after  dropping  accents) 

showing  that  the  locus  of  the  poles  of  all  lines  through  the  point 
(a?i,  y,)  is  the  polar  of  (a?!,  yj. 

PROBLEMS. 

1.  Find  the  polars  of  the  following  points   with  respect   to 

(1,3),  (—2,4),  {k,—p),  (0,  G^),  (6,  sin  A:),  (c,  :^) . 

2.  Find  the  poles  of  the  following  lines  with  respect  to 

^2  _|_  I/'  =  10. 
3a?  — 22/  +  4  =  0  ^^V_^^ 

y  =  mx  +  &  X  —  ^  =  0 

7/=4 

ja?  =  3  +  2r 
( ?/  =  5-f-6r 

3.  Find  the  general  equation  of  the  polar  of  a  point  on  the  J 
axis  with  respect  to  the  circle 

^'  +  2/'  =  r"" ; 

of  a  point  on  the  line  a?  =  2;  on  the  line  x  =  Zy\  on  the  line 
a?  =  3?/  +  2. 

4.  Given  two  diameters  at  right  angles  to  each  other  show  thnt 
the  polars  of  all  points  on  one  are  parallel  to  the  othpr.  a^d 
conversely  that  the  poles  of  all  lines  parallel  to  one  lie  on  the 
other. 

5.  Show  that  if  the  point  (x^,  y^)  lies  on  its  own  polar  with 
respect  to 

it  lies  also  on  the  circle. 


Analytic  Geometry.  73 

6.  Show  that  the  condition  which  must  be  satisfied  in  order 
that  [x\,  i/i)  may  lie  on  the  polar  of  {x^,  ^2)  is  identical  with  the 
condition  which  must  be  satisfied  in  order  that  {X2,  2/2)  niay  lie 
on  the  polar  of  {x^,  1/1),  and  thus  prove  the  following  theorem. 

Let  A  and  B  be  two  points.  Then  if  A  lies  on  the  polar  of  B,  B 
lies  on  the  polar  of  A. 

7.  On  the  basis  of  the  theorem  just  stated  deduce  a  method  of 
constructing  with  ruler  and  compass  the  pole  of  a  given  line 
which  does  not  meet  the  circle  in  real  points. 

8.  Construct  with  ruler  and  compass  the  polar  of  a  point  inside 
the  circle. 

9.  Show  both  geometricly  and  algebraicly  that  the  polar  of  the 
center  of  the  circle  is  the  line  at  infinity. 


CHAPTER  XI. 

ADDITIONAL  WORK  ON  THE  SUBJECT  OF  LOCI. 

57.  Problems  in  which  the  restrictions  on  the 

GENERAL  REMARKS  movement  of  the  tracing  point  are  given  and 
ON  LOCI  PROBLEMS,  the  equation  of  the  locus  demanded  are  ail 
alike  in  the  fact  that  the  method  of  solution  consists 
merely  in  the  translation  of  the  law  of  movement  of 
the  tracing  point  into  algebraic  language.  They  may,  how- 
ever, be  divided  into  two  general  classes.  In  the  first 
class  fall  problems  of  the  type  discussed  in  paragraph  24,  in  which 
the  statement  of  the  law  gives  the  locus  immediately.  In  the 
second  class  fall  problems  of  the  type  discussed  in  paragraphs 
55  and  56,  in  which  the  attempt  to  translate  the  law  of  movement 
of  the  point  leads  to  relations  connecting  the  co-ordinates  of  the 
tracing  point  with  each  other  and  with  certain  arbitrary  para- 
meters. 

In  every  legitimate  locus  problem  in  plane  geometry  the  number 
of  equations  expressing  such  relations,  either  between  the  vari- 
able co-ordinates  and  the  parameters,  or  between  the  parameters 
themselves,  is  always  one  more  than  the  number  of  the  parameters, 
so  that  it  is  possible  by  the  elimination  of  the  parameters  to 
deduce  a  single  relation  connecting  the  co-ordinates  of  the  tracing 
point  with  each  other  and  with  know  n  constants,  i.  e.,  the  equation 
of  the  locus.  If  in  any  particular  case  the  number  of  equations 
is  less  than  this,  one  of  two  things  must  be  true.  Either  the 
conditions  laid  down  do  not  force  the  tracing  point  to  follow  a 
definite  path,  or  the  student  has  failed  to  impose  on  the  co-ordi- 
nates of  the  point  or  on  the  parameters  all  of  the  limitations 
imposed  by  the  problem. 

As  illustrations  of  what  has  been  said  above  consider  the  solu- 
tions of  the  following : 

1.  A  line  of  fixed  length  slides  along  the  co-ordinate  axes,  keep- 
ing one  end  on  each  axis.    Find  the  locus  of  its  middle  point. 


Analytic  Geometry. 


75 


X  = 


The  point  P,  (x,  y) ,  has  its  position 
determined  by  the  two  variable  quanti- 
ties a  and  h.  If  k  is  the  fixed  length  of 
the  line  we  have  as  the  algebraic  trans- 
lation of  the  restrictions  on  the  move- 
ment of  P  the  following  equations : 

_a^        __.])_ 

2  '  ^        2  ^  

a'  +  h'  =  k\ 
From   these  three  equations   we  elim- 
inate the  two  parameters  a  and  1),  and 
deduce  the  single  relation 

the  equation  of  the  desired  locus. 

2.  Given  a  fixed  point  on  a  circle  and  a  variable  chord  through 
that  point.     Find  the  locus  of  the  point  which  divides  the  chord 


Fig.  22. 


in  the  ratio 


Let  0  be  the  fixed  point,  OP  any 
position  of  the  variable  chord,  and 
Q  the  point  whose  locus  we  desire 
to  find.  We  are  free  to  locate  our 
axes  in  any  position,  and  in  order 
that  the  work  may  be  as  simple  as 
possible  we  select  the  radius  of  the 
circle  through  0  as  the  X  axis  and 
the  tangent  at  0  as  the  Y  axis. 
The  circle. then  has  (r,  0)  as  its 
center  and  its  equation  is 

x~  —  2rx  -|-  t/^  =  0. 
The  variation   of  Q   is  evidently 

produced  by  the  variation  of  P  along  the  circle.  Let  P  be  (a?i,  y^) 
and  Q  {x  ,  ?/0  and  we  have  at  once  as  the  translation  into  alge- 
braic language  of  the  limitations  on  the  movement  of  Q 


Fig.  23. 


mx. 


WVy 


m.  +  n  m  -h  n 

two  relations  embracing  x ,  y   and  the  two  variable  parameters 
x^  and  ?/i,  one  relation  less  than  we  need.  It  is  evident  that  the 


76  Analytic  Geometry. 

limitations  force  Q  to  trace  a  definite  locus,  we  must  therefore 
have  failed  to  impose  one  of  the  limitations  of  the  problem. 
Searching  for  this  omitted  limitation  we  soon  see  that  we  have 
not  restricted  (^i,  v/j)  to  the  circle.  Imposing  this  restriction 
we  have  our  desired  third  relation 

which  combined  with  the  two  already  found  enables  us  to  elimi- 
nate m  and  n  and  deduce  the  desired  relation  between  x  and  y\ 


or,  dropping  accents, 


PROBLEMS. 

1.  A,  B,  C  are  three  fixed  points  on  a  straight  line  and  P  a 
variable  point  subject  to  the  condition 

angle  APB  =  angle  BPC. 

Find  the  locus  of  P. 
(In  this  as  well  as  the  other  problems  of  this  list  the  student 
will  find  it  well  to  locate  his  axes  in  such  a  way  as  to  give  tlie 
greatest  possible  simplicity  to  the  work  without  destroying  the 
generality  of  the  problem.) 

2.  Through  a  fixed  point  0  on  a  circle  chords  are  drawn  and 
on  each  chord,  extended,  a  point  P  is  taken  such  that  OP  is  twice 
the  length  of  the  chord.    Find  the  locus  of  P. 

3.  Subject  the  general  circle 

{x  —  a)^-{-{y  —  'b)^  =  r- 

to  the  condition  of  passing  throuirh  the  two  fixed  points  (a?,,  iy, ) 
and  (Xo,  ifo)  and  find  the  locus  of  the  center. 

4.  Find  the  locus  of  the  point  from  which  a  fixed  segment  AB 
on  a  ffiven  line  subtends  a  right  angle. 

5.  The  two  tangents  from  ft  variable  point  to  a  fixed  cir^'lp 
make  with  each  other  a  constant  angle.  Find  the  locus  of  the 
variable  point. 

6.  The  distance  of  the  point  P  from  a  fixed  point  on  a  circle 
centered  at  the  origin  is  equnl  to  the  slope  of  the  polar  of  P  with 
respect  to  the  circle.    Find  the  locus  of  P 

7.  A  and  B  are  two  fixed  points.  The  rlistnnce  of  P  from  A 
equals  the  cosine  of  the  angle  PAB.    Find  the  locus  of  P. 


Analytic  Geometry.  77 

8.  The  distance  of  P  from  its  polar  with  respect  to  a  given 
circle  centered  at  the  origin  is  equal  to  the  slope  of  the  polar. 
Find  the  locus  of  P. 

9:  A  line  moves  parallel  to  its  original  position.  On  the  line 
a  point  P  is  taken  so  that  the  distance  from  P  to  the  Y  axis  is 
equal  to  the  distance  from  P  to  the  point  where  the  line  meets  the 
X  axis.    Find  the  locus  of  P. 

10.  A  line  of  constant  length  slides  on  the  co-ordinate  axes, 
keeping  one  extremity  on  each  axis.  Find  the  locus  of  its  pole 
with  respect  to  a  given  circle  centered  at  the  origin. 

11.  The  distance  of  the  point  P  from  its  polar  with  respect  to  a 
fixed  circle  centered  at  the  origin  is  equal  to  the  sum  of  the  inter- 
cepts of  the  polar  on  the  co-ordinate  axes.    Find  the  locus  of  P. 

12.  A  is  a  fixed  point  outside  and  Q  a  variable  point  on  the 
circumference  of  a  fixed  circle.  Find  the  locus  of  the  point  ou 
the  line  AQ  whose  distance  ratio  with  respect  to  A  and  Q  is  3. 

13.  Find  the  locus  of  the  poles,  with  respect  to  a  given  circle, 
of  a  system  of  parallel  straight  lines. 

14.  A  variable  line  is  subject  to  the  condition  that  it  must  be 
tangent  to 

a?2  _|_  ^2  ^  16. 

Find  the  locus  of  its  pole  with  respect  to 

a?2  +  2/'  =  4. 

15.  The  line  joining  the  point  P  to  a  fixed  point  on  the  circum- 
ference of  a  given  circle  is  perpendicular  to  the  polar  of  P 
with  respect  to  the  same  circle.    Find  the  locus  of  P. 

16.  Which  of  the  loci  deduced  above  are  circles? 


CHAPTER  XIII. 
THE  GENERAL  EQUATION  OF  THE  SECOND  DEGREE. 

58.  We  take  up  now  the  study  of  the  general 
NATURE  OF  THE  conic  as  represented  by  the  general  equation 
THE^METHOD°  ^^  ^^  ^^^  second  degree,  a  special  ease  of  which 
EMPLOYED.  ^^^  heen  studied  in  the  chapter  on  the  circle. 

We  shall  show  that  every  conic  has  two  axes 
of  symmetry  and  that  by  making  the  axes  of  co-ordinates  coinci- 
dent with  or  parallel  to  these  axes  of  symmetry  all  equations  of 
the  second  degree  are  reduced  to  three  type  forms.  These  type 
forms  will  then  be  investigated  in  much  the  same  manner  as  the 
circle  was  investigated  in  Chapter  XI. 

For  the  first  part  of  the  investigation  we  shall  use  the  para- 
metric form  of  the  equations  of  a  straight  line,  developed  in  para- 
graph 39,  which  bring  into  evidence  a  fixed  point  (a?i,  y^),  the 
direction  of  a  line  through  the  point,  and  the  distance  measured 
along  the  line  from  the  fixed  point  to  a  variable  point.  By  put- 
ting the  variable  point  on  the  conic  and  rotating  the  line  we 
shall  be  able  to  investigate  the  curve  by  noting  the  changing  value 
of  the  distance  from  the  fixed  point  to  the  variable  point  on  the 
conic,  in  much  the  same  manner  that  the  bottom  of  a  lake  is  investi- 
gated by  measurement  of  its  depth  at  various  points. 

59.  Consider  any  point  (x-^,  i/J  and  write  anv 
THE  "r"  EQUATION,    jj^^  through  it  in  the  form 

a?  =  a?i  +  ^^ 

where  I  and  m  are  the  cosines  of  the  angles  which  the  line  makes 

with  the  X  and  Y  axes.    To  find  the  distances  along  the  line  from 

the  point  (a?i,  y^)  to  the  conic  we  substitute  x  and  y,  as  given  by 

the  equations  of  the  line,  in  the  general  equation  of  the  second 

degree 

ax^  +  hy^  +  2hxy  +  2gx  +  2f^  +  c  =  0. 


Analytic  Geometry.  79 

This  gives  us 

( ax^'  +  ly^^  +  2hx,y,  +  2gx,  +  2fy,  +  c) 

+  2r\nax,-^}iy,-^g)-{-m{lix^-\-ly,  +  t)] 

an  equation  in  r  whose  roots  are  the  distances  from  the  fixed  point 
to  the  conic.  Since  the  equation  is  a  quadratic  there  are  two  such 
distances  and  we  have  the  theorem : 

A  conic  is  met  by   any  straight  line  in  two  points. 

60.  If  the  point  (x^^,  y^)  is  so  placed  as  to  be 
ONE  CHORD  IS  midway  between  the  points  of  intersection 
BISECTED  AT  ANY  of  the  line  and  the  conic  the  two  values  of  / 
POINT.  given  by  the  r  equation  are  equal  in  value 
and  opposite  in  sign.  The  necessary  and  sufficient  condition  for 
this  is  the  vanishing  of  the  coefficient  of  the  first  degree  term  in  r, 
i.  e., 

l{ax^  +  hy^  +  g)  +  m{hx^  +  hy^  +  f  )=  0.  A 

This  equation  maybe  satisfied  in  several  ways.  First,  a?i  and  ?/imay 
be  given  any  arbitrary  values  and  the  equation  satisfied  by  proper 

choice  of  — ,  i.  e.,  by  giving  the  proper  direction  to  the  line.    Note 

that  only  one  value  of    —   will  satisfy  the  equation.     From  this 

method  of  satisfying  the  condition  we  derive  the  theorem : 

Through  any  point  in  the  plane  there  may  be  drawn  one  and  Jn 
general  but  one  chord  of  a  given  conic  which  is  bisected  at  that 
point. 

61.  Again,  it  is  always  possible  to  find  one  and 
CENTER  OF  A  CONIC,  only  one  point  which  satisfies  both  of  the 

equations 

ax  +  hy+g  =  0 
Jix  +  'by+f  =  0 

If  this  point  is  taken  as  {Xj^,  y-^)  the  coefficient  of  r  is  zero  without 

regard  to  the  value  of  — ^  i.  e.,  the  chord  is  bisected  at  {x^,  y-^) 

no  matter  what  its  direction.    This  gives  us  the  second  theorem : 
Every  conic  has  a  center  of  symmetry  whose  co-ordinates  are 


80  Analytic  Geometry. 

determined  by  the  equations 

62.  Again,  let     —   have  a  fixed  value.     It  is 

possible  to  satisfy  the  condition  A  by  choice 
CONIC.  ^  _ 

of   (ii?i,  2/1)  •     I^  f^ct  if  we  regard    —     as 

c 

fixed  and  x^  and  y-^  as  variable  the  condition  becomes  the  equa- 
tion   of   a   straight    line,  which    is   evidently   the   locus   of   the 

middle  points  of  a  system  of  parallel  chords  with  the  slope  — . 

Such  a  locus  is  called  a  diameter  of  the  conic.  It  is  easy  to 
see  that  all  diameters  pass  through  the  center  of  the  conic.  We 
have  now  our  third  theorem : 

The  equation  of  the  diameter  bisecting  the  family  of  chords 

whose  slope  is   —  is 

l(ax  +  hy+g)  +  m{lix+'by  +  f)=^ 

63.  An  axis  of  symmetry  differs  from  other 
AXES  OF  SYMMETRY  diameters  in  that  it  is  perpendicular  to  the 
OF  A  CONIC.                chords  it  bisects.    Given  a  family  of  chords 

of  slope  —  the  corresponding  diameter  has  the  slope 

L 

al  -\-  hm 


hi  +  fe^ 

» 

If  this  diameter  is  to  be  perpendicular  to  the  chords  it  bisects  we 
must  have 

'm\  / al  +  hm\ -j 

l)\      hl^lmj 
i.  e.,  aim  +  ^^^  =  hP  -{-Mm 


or 


/m\  \  /a  —  ^\  m 


The  roots  of  this  equation  are  the  slopes  of  systems  of  chords 
perpendicular  to  the  diameter  which  bisects  them.  The  equation 
is  a  quadratic,  therefore  there  are  two  such  systems.  The  product 
of  the  two  roots  is  — 1,  therefore  the  two  systems  are  perpen- 
dicular to  each  other.    We  have  now  our  fourth  theorem : 

Every  conic  has  two  axes  of  symmetry,  and  these  two  axes  are 
perpendicular  to  each  other. 


Analytic  Geometry. 


81 


64.  The  work  of  the  last  article  enables  us  to 

REDUCTION  OF  THE  determine  the  angles  which  the  two  axes  of 
GENERAL  EQUATION. symmetry  make  with  the  X  axis.  Let  us 
assume  that  the  axes  of  co-ordinates  have 
been  made  parallel  to  the  axes  of  symmetry  by  rotation  through 
one  of  these  angles,  and  let  the  form  of  the  equation  referred  to 
these  new  axes  be 

Ax^  +  Btf  +  2Hwij  +  2Gx  +  2Fy  +  C  =  0.  (1) 

Two  questions  are  now  before  us.  This  particular  choice  of  axes 
must  entail  certain  values  for  some  of  the  coefficients  of  the  equa- 
tion or,  what  amounts  to  the  same  thing,  special  relations  between 
them.  The  determination  of  these  values  or  relations  is  our  first 
question,  and  the  determination  of  the  distance  from  the  axes  of 
co-ordinates  to  the  axes  of  symmetry  is  the  second. 

Let  AB  and  CD  (Fig.  24)  be  the 
axes  of  symmetry  and  let  their  dis- 
tances from  the  axes  of  co-ordinates 
be  k  and  I  (equations  y  — 1  =  0, 
X  —  /c  =  0 ) .  The  necessary  and  suffi- 
cient condition  that 

a?  — A;  =  0 

may  be  an  axis  of  symmetry  is  that 

if  any  point  P^,  co-ordinates  ( Ic  +  r, 

2/i),  be  on  the  curve,  the  point  Q,  co-  ^^^-  ^*- 

ordinates   (k  —  r,  y^),  shall  also  be  on  the  curve,  i.  e.,  if 

A  ( k+r)  ^-j-By,^-\-2n  ( k+r)  y^+2G-  ( A;+r )  -{-2Fy^-\-C-=0 
so  also  Aik~r)^--\-By^^+2H{k—r)y^-^2a{k—r)-\-2Fy^-\-G=0 

Subtracting  the  second  from  the  first  and  dividing  by  4r  (Wha^ 
right  have  we  to  divide  by  r?)  we  have 

Ak-\rG  +  Hy^  =  0 

an  equation  of  the  first  degree  in  t/i  which  is  satisfied  by  the  // 
co-ordinate  of  any  point  on  the  conic.  This  can  be  true  in  only 
two  ways ;  either  the  general  conic  must  consist  of  straight  lines, 
or  the  equation  last  written  is  not  a  condition  but  an  identity. 
If  the  former  of  these  alternatives  were  true  the  general  equatiou 
must  split  up  into  first  degree  factors,  but  it  does  not;  we  have 

therefore  Ak  -4-  G  +  Hp^  =  0 

i.e.,  Ak  +  G  =  0     and     ^'^O. 


82  Analytic  Geometry. 

Similarly  treating  the  other  axis  of  symmetry  we  have 
Bl-\-F  =  0     and     H=0. 

If  then  the  axes  of  co-ordinates  are  taken  parallel  to  the  axes 
of  symmetry  of  the  conic  the  equation  has  no  term  in  xy,  and  the 
coefficients  A,  B^  F,  G,  are  connected  with  each  other  and  with 
the  distances  from  the  axes  of  co-ordinates  to  the  axes  of  symmetry 
by  the  relations 

(The  point  [h,  I)  is  evidently  the  ceilter  of  S3'mmetry,  and  its  co- 
ordinates might  therefore  have  been  found  by  the  method  of  article 
61.) 

Apply  the  transformation 

x  =  x—  ^ 
A 

F 

and  the  new  axes  of  co-ordinates  coincide  with  the  axes  of  sym- 
metry of  the  conic,  while  the  equation  reduces  to  the  form 

(1)  Ax^  +  Bif  +  K  =  () 

in  which  K  denotes  the  new  constant  term.  If  either  A  or  ^  is 
zero  (both  cannot  be,  problem  4,  article  33)  the  above  trans- 
formation cannot  be  made,  since  in  this  case  the  center  of  sym- 
metry and  one  of  the  axes  of  symmetry  are  at  infinity.  The 
simplification  of 

Ax^  +  Btf  +  2Gx  +  2Fy  +  C  =  0 

must  therefore  in  this  case  be  accomplished  in  some  other  way. 
Let  A  be  the  coefficient  that  vanishes.  There  is  nothing  to  prevent 
our  applying  the  transformation 

'       F 

which  makes  the  X  axis  coincide  with  that  axis  of  symmetry  which 


*i.  e.,  the  special  choice  of  axes  made  at  the  beginning  of  this  paragraph 
leaves  ( 1 ) ,  not  in  the  general  form  there  written ,  but  in  the  special  form 

Aa?2  4-  Bi/2  —  2kAx  —  2lBy  4-0=0. 


Analytic  Geometry.  83 

is  in  the  finite  part  of  the  plane.    This  transformation  reduces  the 
equation  to  tlie  form 

By^  +  2Gx  +  L  =  0 
where  L  is  the  new  constant  term.    This  curve  crosses  the  X  axis 
at  the  point  ( — ,  0)  which  is  in  the  finite  part  of  the  plane 

so  long  as  G  does  not  vanish.    We  therefore  apply  the  transforma- 
tion 

2G 
which  moves  the  Y  axis  to  'this  point  of  intersection  and  reduces 
the  equation  to  the  form 

(2)  By^  +  2Gx  =  0. 

If  G  vanishes  the  equation 

By'~  +  2Gx  -\-L  =  0 
reduces  at  once  without  transformation  to 

(3)  By'  +  L  =  0. 

We  have  now  succeeded  in  showing  that  each  and  every  equation 
of  the  second  degree  in  two  variables  may,  by  a  mere  transforma- 
tion of  co-ordinates,  be  reduced  to  one  of  the  tj^pes  (1),  (2),  (3) 
above.     (3)  reduces  at  once  to 


^l■ 


z 


representing  two  real  or  imaginary  lines  parallel  to  the  X  axis.. 
We  may  therefore  dismiss  it  from  further  consideration.  (1)  and 
(2)  remain  to  be  studied.  Since  all  the  conies  reducible  to  (1> 
have  their  centers  in  the  finite  part  of  the  plane  we  shall  fre- 
quently refer  to  them  as  central  conies. 


CHAPTER  XIV. 

THE  ELLIPSE  AND  THE  HYPERBOLA. 

65.  We  take  up  now  the  consideration  of  type 

DETERMINATION  OF  (1)   of  the  preceding  chapter,  the  equation 
^^^^'  Ax^  +  Btf  +  K=^0. 

If  Evanishes  the  equation  at  once  reduces  to  the  form 


und  therefore  represents  two  real  or  imaginary  straight  lines 
through  the  origin.  If  K  does  not  vanish  and  A,  B,  and  K  have  all 
the  same  sign,  the  equation  cannot  be  satisfied  by  any  real  poiot 
{sum  of  three  positive  or  three  negative  quantities  cannot  be  zero)  ; 
and  since  we  are  for  the  present  interested  only  in  real  loci  we 
«hall  give  no  further  attention  to  this  case.    When  the  signs  are 

not  all  the  same  divide  by  K^  put  —  ^  =  a, =  p  and  the 

A  K 

equation  reduces  to  the  form 

This  curve  meets  the  X  axis  in  the  points  (  -^,  0)  and  ( — ,  0) 

and  the  Y  axis  in  the  points  (0,  -^)  and  (0, -z),  and  the  lengths 

•     •  yp  V P 

t)f  the  segments  determined  by  the  curve  on  the  X  and  Y  axes 

(and  hence  on  the  axes  of  symmetry)  are    -^^    and    -^.     These 

l/a  1//8 

values  might  logically  be  called  the  lengths  of  the  axes  of  the 
curve,  but  as  one  or  the  other  of  the  quantities  a,  (3  may  be  neg- 
ative mathematicians  have  agreed  to  define  the  lengths  of  the  axes 

(  2 


as  the  moduli  (see  appendix  E)  of  ttiese  values,  i.  e., 


1    a 


VP 


and  in  this  way  avoid  the  introduction  of  imaginary  lengths.     If 
we  denote  the  semi-axes  thus  defined  by  a  and  h  we  have 


2a  = 


2 


1/  a 


»HA 


Analytic  Geometry. 


85 


If  a  and  (^  are  both  positive 
we  have 

a  0 

and  the  equation  takes  the  form 

a'  If 
Solve  this  equation  for  y  in 
terms  of  £i?  and  the  truth  of  the 
following  statements  is  at  once 
evident.  As  x  increases  in  num- 
erical value,  or,  to  say  the  same 
thing  more  technically,  as  |a?|  in- 
creases, \y\  decreases  from  the 

y 


Fig.  25. 

value  5  which  it  has  when  \x\ 
is  zero  to  the  value  zero  which  it 
has  when  |a?|  is  a.  As  |a?|  in- 
creases beyond  a,  \y\  increases 
indefinitely,  but  since  y  is  imag- 
inary for  these  values  the  corre- 
sponding points  are  not  repre- 
sented in  the  plane.  The  curve 
therefore  lies  wholly  within  the 
rectangle  formed  by  the  lines 

X  ±a^^  and  i/  ±  &  =  0 
Careful  plotting  will  show  it  to 
be  of  the  form  here  given.     It 
is  called  an  ellipse. 


If  one,  let  us  say  ^  of  the 
quantities  a,  /?  is  negative,  we 
have 


-P 


1)' 


and  the  equation  takes  the  form 


1j' 


Solve  this  equation  for  y  in 
terms  of  x  and  the  truth  of  the 
following  statements  is  at  once 
evident.  As  \x\  increases,  be- 
ginning at  zero,  \y\  decreases 
from  the  value  1),  which  it  has 
when  \x\  is  zero,  to  the  value 
zero,  which  it  has  when  \x\  is  a. 

Y 


Fig.  26. 

Over  this  range,  however,  y  is 
imaginary  so  the  corresponding 
points  are  not  represented  in 
the  plane.  As  |^|  increases  be- 
yond a,  \y\  increases  indefin- 
itely. The  curve  therefore  lies 
wholly  without  the  lines 

x±  a  =  0 

and  extends  upward  and  down- 
ward indefinitely.  Careful  plot- 
ting will  show  it  to  be  of  the 
form  here  given.  It  is  called  an 
hyperbola. 


86 


Analytic  Geometry. 


In  plotting  this  diagram  a 
was  assumed  greater  than  I). 
If  a  is  less  than  h,  the  conic  is 
turned  along  the  other  axis. 
The  question  of  size  is  evidently 
the  important  one  in  distin- 
guishing between  the  axes  and 
they  are  therefore  spoken  of  as 
major  and  minor.  In  the  devel- 
opment of  the  theory  of  the 
ellipse  we  shall  assume  that  a 
denotes  the  length  of  the  semi- 
major  axis. 


In  plotting  this  diagram  ^ 
was  assumed  negative.  If  a 
is  negative  the  conic  is  turned 
along  the  other  axis.  The  im- 
portant question  here  is  not  one 
of  size  but  of  the  character  (real 
or  imaginary)  of  the  points  in 
which  the  conic  meets  the  axes. 
The  axis  met  by  the  conic  in 
real  points  is  called  the  trans- 
verse axis,  the  one  met  in  im- 
aginary points  is  called  the 
conjugate  axis.  In  the  devel- 
opment of  the  theory  of  the 
hyperbola  we  shall  assume  that 
a  denotes  the  length  of  the 
semi-transverse  axis. 


66. 


EARLY  GEOMETRIC 
DEFINITIONS. 


These  curves  were  well  known  to  geometri- 
cians before  the  invention  of  analytic  geom- 
etry, and  each  of  them  had  its  geometric  defi- 
nitions.   Two  of  these  are  as  follows : 
(A)  An  ellipse  I  An  hyperbola 

is  the  locus  of  a  point  the 


sum 


difference 


IS 


of  whose  distances  from  two  fixed  points,   called  the   foci 

constant. 

(B)  An  ellipse  |  An  hyperbola 

is  the  locus  of  a  point  whose  distance  from  a  fixed  point  divided  by 

its  distance  from  a  fixed  line  is  a  constant 

less  I  greater 

than  unity. 

The  fixed  point  is  called  the  focus,  the  fixed  line  the  directrix,  and 

the  constant  ratio  the  eccentricity. 

If  we  attempt  to  deduce  the  equations  of  these  curves  directly 
from  the  definitions  just  given  the  resulting  forms  will  depend 
upon  the  choice  of  co-ordinate  axes.  In  deducing  the  equations 
from  definition  (A)  we  choose  the  line  joining  the  two  foci  as  the 
X  axis  and  the  perpendicular  bisector  of  the  segment  between  the 


Analytic  Geometry. 


87 


foci  as  the  Y  axis.  Then  if  the  distance  between  the  foci  be  taken 
as  2c  the  foci  are  ( — c^  0)  and  (c_,  0)  and  our  definitions  lead  at 
once  to  the  equations 


i/  («  +  cY  +  if 


+  }/  {x—cY  +  i/  =  2k 


V  (^  +  cY  +  V 


—V{x  —  GY-\-'if=2k 


where  2h  is  the  constant 
sum  I  difference 

mentioned  in  the  definition.    Rationalize  and  reduce  and  each  of 
the  forms  leads  to  the  single  equation 

,2  ,2 


X 


+ 


If  we  compare  this  with  our  standard  forms  we  have 


l)^  =  h' 


=  a^ 


a^  —  l^ 


k^ 

r,2 


c^  =  a^-\-J)' 


and  we  have  as  foci  the  two  real  points 


(Va^-6^0) 


(-^a^^V~,0) 


Note  that  the  foci  were  assumed  on  the  X  axis  and  the  resulting 
equation  identified  with  one  for  which  the  X  axis  is  the 

major  I  transverse 

axis  of  symmetry.    Hence  we  may  say  that  the  foci  are  two  real 
points  on  the 

major  j  transverse 

axis  of  the  conic  and  symmetrically  situated  with  respect  to  the 
other  axis. 

li  on  the  other  hand,  the  foci  are  assumed  on  the  Y  axis  the 
resulting  equation  is 

2  ^,2 

+ 


k'  —  c'      k' 
Identifying  this  with  the  same  equations  as  before,  we  have 


l)^  =  k^ 
a^  =  k^ 


—  h^  =  k^ 
a^  =  k^  —  c^ 

=  — Z>2  — c^ 
c^  =  —  a^—h' 


88  Analytic  Geometry. 

and  we  have  as  foci  the  two  imaginary  points 


(0,  V^'-a^) 


(0,  — V6'  — a') 


(0,  v-«^-&^) 


(0,  —V— «'  —  &') 


Note  that  in  this  ease  the  foci  are  assumed  on  the  Y  axis  and  the 
resulting  equation  identified  with  one  for  which  the  Y  axis  is  the 

minor  |  conjugate 

axis  of  symmetry.    Hence  we  may  say  that  the  foci  are  two  imagin- 
ary points  on  the 

minor  |  conjugate 

axis  of  the  conic  and  symmetrically  situated  with  respect  to  the 
other  axis. 

But  since  either  assumption  leads,  when  c  and  k  are  properly 
determined,  to  the  same  equation 


nl- 


.2     -      J. 


a 

it  follows  that  the  conic  represented  by  this  equation  (i.  e.,  any 
central  conic)  has  four  foci,  two  real  on  the 

major  I  transverse 

axis,  and  two  imaginary  on  the 

minor  |  conjugate 

axis.    Hereafter  when  the  foci  are  referred  to  it  is  understood  that 
the  reference  is  to  the  real  foci  unless  both  are  mentioned. 

In  deducing  the  equations  from  definition  (B)  we  choose  the 
directrix  as  the  Y  axis  and  the  perpendicular  let  fall  upon  it 
from  the  focus  as  the  X  axis.  We  denote  the  distance  from  the 
focus  to  the  directrix  by  d  and  the  eccentricity  by  e..  The  definition 
then  leads  at  once  to  the  equation 


w 
or  on  reduction 

3c^(l~e^)  +  if  —  2dx  +  d^  =  0. 

The  absence  of  the  xy  term  shows  that  the  axes  of  co-ordinates  are 
parallel  to  the  axes  of  symmetry,  and  the  absence  of  the  y  term 
shows  that  the  X  axis  coincides  with  an  axis  of  symmetry.  But 
the  presence  of  the  x  term  shows  that  the  Y  axis  does  not  coincide 
with  an  axis  of  svmraetrv  and  a  transformation  of  co-ordinates 


Analytic  Geometry. 


80 


must  be  made  before  we  can  compare  our  equation  with  the  stand- 
ard forms.  We  therefore  bring  the  Y  axis  into  coincidence  with 
the  other  axis  of  symmetry  by  the  substitution 


w 


■  x'-i- 


where 


d 


is  the  distance  from  the  directrix  to  the  axis  of 


1-e' 

symmetry  to  which  it  is  parallel.     The  transformation  gives  an 
equation  which  reduces  finally  to 


72     2 

a  e 


+ 


V 


=  1. 


a  —  e'f         l~e' 
an  equation  of  the  desired  standard  form 

For  the  ellipse  e  is  less  than 
unity  and  the  coefficients  of 
both  w^  and  y^  are  positive. 
Comparing  our  present  equa- 
tion with  the  standard  form  we 
have 

dV 


h'  = 


(1  —  e'] 
dV 


whence 


=  1 


Va' 


Again,  replacing  e  by  —  and 


e^  by    ^  we  have,  on  solv- 
a 


ing  for  d 


d=± 


For  the  hyperbola  e  is  greater 
than  unity  and  the  coefficient 
of  w^  is  positive  while  that  of 
y^  is  negative.  Comparing  our 
present  equation  with  the 
standard  form  we  have 


a  = 


dV 


(1-^7 


—  h 


2  _  d.V 

(1  -  e') 


whence 


e 

a 

e 
a 

Again, 

replacing 

e  b.y 

c 
a 

and 

1  —e' 

by  — 

a 

we 

have 

on 

solving 

for  d 
d  = 

-  ^ 

c 

The  quantity  c  will  hereafter  be  called  the  linear  eccentricity. 


90  Analytic  Geometry. 

An  investigation  of  the  significance  of  the  double  sign  of  d 
leads  to  interesting  results. 

To  take  the  positive  sign  is  to  assume  that  the  focus  is  on  the 
right  of  the  directrix,  while  the  distance  which  the  Y  axis  must  be 
moved  to  pass  from  coincidence  with  the  directrix  to  coincidence 
with  the  axis  of  symmetry  is 

positive  (e  less  than  unity)  and  negative  (e  greater  than  unity), 
greater  than  d. 

To  take  the  negative  sign  is  to  assume  that  the  focus  is  on  the 
left  of  the  directrix,  while  the  distance  through  which  the  Y  axis 
Is  moved  is 

negative  |  positive 

with  the  same  numerical  value  as  before. 

In  other  words,  if  we  start  with  a  focus  on  the  right  of  a 
directrix  and  move  the  Y  axis  a  certain  distance  to  the 

right  I  left 

we  reduce  the  equation  to  a  certain  form.  If  we  start  with  a 
focus  on  the  left  of  a  directrix,  we  have  a  different  set  of  axes 
and  a  different  equation.  But  when  we  move  the  Y  axis  the 
same  distance  as  before  to  the 

left  I  right 

we  reduce  the  equation  to  the  same  form  as  before,  i.  e.,  the 
two  equations  given  by  the  two  signs  of  d  represent-  the  same 
curve,  but  referred  to  different  sj-stems  of  co-ordinates. 

Evidently  therefore  the  two  signs  of  d  correspond  to  two  foci 
and  two  directrices.  Evidently  also  the  directrices  are  sym- 
metrically situated  with  respect  to  one  axis  of  symmetry  of  the 
conic  and  cross  the  other  axis  at  points 

without  I  within 

the  segment  determined  on  that  axis  by  the  foci.  It  is  not  difficult 
to  show  that  these  points  are  also 

without  I  within 

the  segment  determined  on  the  axis  by  the  intersection  of  the 
axis  and  the  conic. 

There  are  of  course  a  pair  of  directrices  corresponding  to  the 
pair  of  imaginary  foci.  These  are  however  imaginary,  a  state- 
ment whose  proof  will  follow  at  once  from  the  solution  of  problem 
8,  article  71. 


Analytic  Geometry. 


91 


67.  Delini- 

MECHANICAL  tion       (A) 

CONSTRUCTIONS.  leads  to  a 
simijle  me- 
chanical method  of  construct- 
ing an  ellipse.  Fasten  at  the 
two  foci  the  two  ends  of  a  cord 
whose  length  is  the  constant 
sum  of  the  focal  distances  of 
the  tracing  point.  Draw  the 
cord  to  one  side  with  a  pencil 
and  draw  the  pencil  along  keep- 


FiG.   27, 


ing  the  cord  tightly  drawn. 
The  resulting  curve  evidently 
satisfies  the  definition  of  an 
ellipse. 


Definition  (A)  leads  to  a 
simple  mechanical  method  of 
constructing  an  hyperbola. 
Make  a  ruler  of  the  form  shown 
in  Fig.  28  with  the  center  of 
the  opening  P  on  the  straight 
edge  AB  extended.  By  means 
of  this  opening  P  pivot  the 
ruler  at  one  focus  F^,  To  the 
other  end,  B,  of  the  ruler  fasten 
one  end  of  a  cord  shorter  than 
the  ruler  by  an  amount  equal 


(^ 


Fig.  28. 

to  the  constant  difference  of 
the  focal  distances  of  the  trac- 
ing point  and  fasten  the  other 
end  of  the  cord  at  the  other 
focus  F^.  Rotate  the  ruler 
about  i*^i,  keeping  the  cord 
pressed  tightly  against  the 
ruler  by  a  pencil.  The  result- 
ing curve  evidently  satisfies 
the  definition  of  an  hyperbola. 
The  other  branch  is  drawn  by 
reversing  the  apparatus. 

PKOBLEMS. 

1.  Let  the  lengths  and  positions  of  the  axes  of  an  ellipse  and 
an  hyperbola  be  given.  Deduce  geometric  constructions  for  the 
foci  and  the  directrices. 


92  Analytic  Geometry. 

2.  Find  the  eccentricity,  lengths  of  axes,  location  of  foci  and 
location  of  directrices  of  the  following  central  conies : 

Zx^  —  2if  +  1  =  0  4fl?2  +2i/2  =12 

—  9a?2  +  102/-  :=  144  4.x''  -\- y'^  =  1 

3.  The  following  conies  are  assumed  to  have  their  centers  at  the 
origin  and  their  axes  of  symmetry  as  axes  of  co-ordinates.  Find 
their  equations. 


(1)  '-t^ 

(3)   d  =  \ 

e  =  4. 

(5)   d  =  2, 

4 

(2)   a=^, 

c  =  2. 

(4)   a  =  % 

1)  =  2. 

(G)   c  =  4, 

a  =  3. 

4.  Express  the  distance  between  the  two  directrices  in  terms  of 
a  and  e  and  show  that  for  the  ellipse  it  is  greater  than  2a^  and  for 
the  hyperbola  less  than  2a. 

5.  Let  the  length  of  the  transverse  axis  of  an  hyperbola  be  con- 
stant and  let  the  eccentricity  increase  indefinitely.  Show  that 
under  these  conditions,  the  directrix  tends  to  coincide  with  the 
conjugate  axis. 

6.  To  what  limiting  form  does  the  ellipse  tend  as  a  tends  to  h  ? 
What  is  the  limit  of  the  eccentricity? 

7.  When  a  tends  to  6  the  hyperbola  tends  to  the  limiting  form 
represented  by 

x^  —  y^  =  -b^ 

which  is  called  an  equilateral  hyperbola.    What  is  its  eccentricity? 

8.  Show  that  as  an  ellipse  tends  to  a  circle  the  distance  of  the 
directrix  from  the  center  tends  to  infinity. 

9.  If  in  any  conic  there  be  erected  at  the  focus  a  perpendicular 
to  that  axis  of  symmetry  which  passes  through  the  focus,  the 
distance  between  the  two  points  in  which  this  perpendicular  meets 
the  curve  is  called  the  length  of  the  parameter  of  the  conic.  This 
length  is  usually  denoted  by  2p.  If  the  conic  is  a  central  conic 
referred  to  its  axes  of  symmetry  as  co-ordinate  axes,  the  parameter 
might  be  defined  as  the  double  ordinate  through  the  focus,  or  as 
that  portion  of  the  line  x-\-  c  =  0  or  x  —  c  =  0  included  between 
its  intersections  with  the  conic.  Show  that  for  the  ellipse  the 
semi-parameter  p  is  a  third  proportional  to  the  semi-axes.  Show 
also  that  p  =  a{l  —  e^). 

10.  Determine  the  corresponding  values  of  p  for  the  hyperbola. 

11.  Move  the  axis  of  Y  so  that  it  shall  become  the  tangent  at 
the  left  hand  vertex  of  the  conic.     (The  vertices  of  a  conic  are  its 


Analytic  Geometry.  93 

intersections  with  its  axes  of  symmetry.)     The  equation  of  tlie 
ellipse  then  reduces  to  the  form 

a 
or  •  'i/  =  2px(l  —  ^). 

Let  a  and  &  increase  indefinitely,  but  in  such  a  way  that  the  ratio 

7/ 

- {=p)   remains  constant.     What  is  the  limiting  value  of  s? 

a 

What  is  the  limiting  form  of  the  second  of  the  two  equations  just 
given  ? 

12.  Make  a  similar  investigation  for  the  hyperbola. 

13.  Compare  the  limiting  forms  developed  in  the  last  two  prob- 
lems with  equation  (2),  article  64,  and  note  that  the  assumptions 
just  made  concerning  a  and  ft  have  the  eft'ect  of  moving  the  center 
of  the  conic  to  infinity,  and  therefore  correspond  to  the  assump- 
tions which  led  to  equation  (2) .  Hence  we  may  state  the  theorem  : 
The  conic  represented  by  form  (2)  is  the  limiting  form  of  both 
the  ellipse  and  the  hyperbola  as  the  eccentricity  tends  to  unity. 

14.  The  lines  joining  any  point  on  a  conic  to  the  foci  are  called 
focal  radii.  Denote  their  lengths  by  r  and  r',  let  the  axes  of  co- 
ordinates be  the  axes  of  symmetry,  and  show  that  for  the  hyperbola 

r  =  ex  —  a,  r=ex-\-a. 

15.  What  are  the  corresponding  values  for  the  ellipse? 

68.  Applying  to  the  special  equation   of  the 

DIAMETERS.  second  degree 

ax^  -f  (3(/  =  1 
the  general  formulae  for  the  equation  of  a  diameter  of  a  coni<i 
developed  in  article  62,  we  find  that  the  equation  of  the  diameter 
bisecting  chords  of  slope  m  is 

ax  -f  f^7ny  =  0  i.e. 

h^x -\- a^my  =  0  I  h-x — a^my  =  0 

Given  two  diameters 

y  =  m^x  and  y  =  m^x 
the  work  just  done  shows  that  the  necessary  and  sufficient  condi- 
tion that  the  second  diameter  shall  bisect  all  chords  parallel  to  the 
first  is 


m^m,,  =  —  ~  i.e., 


m^m^  =  —  ^ 
a 


^1^2  =  -T 


94  Analytic  Geometry. 

From  the  form  of  this  condition  it  is  evident  that  if  the  second 
diameter  bisects  all  chords  parallel  to  the  first,  the  first  also 
bisects  all   chords  parallel  to  the   second.       Such   a   pair   of 
diameters  are  said  to  be  conjugate  diameters. 
Let 

y  =  m^x  and  y  =^  m^x 

be  a  pair  of  conjugate  diameters  of  the  conic 

and  let  the  points  in  which  they  meet  the  conic  be  (a?i,  t/i)  ?  (^2?  2/2)  > 
(— a?i,  — 2/i)\  (— a?2,  — 1/2)  then 

m-^  =  ^—  and  m^  =  -^- 
and  therefore  since  the  diameters  are  conjugate 

M?  =  _-?  (1) 

Also  since  the  points  are  on  the  conic 

ax,'  +  %/  =  1  (2) 

and  ax,'  +  ^i/.^  =  1  (3) 

Solve  (2)  and  (3)  for  x^  and  y^  and  substitute  in  (1)  whence  we 
have  

Substitute  this  value  of  a?2  in  (1)  and  we  have 

If  expressed  in  terms  of  the  lengths  of  the  axes  this  becomes 


^2  =  ±  f-7/i, 
h 

2/2  ^^  =P      ^ij 
a 


0 


i/2  —  -t-  *       '^i? 
a 


where  the  upper  sign  of  x  is  paired  with  the  upper  sign  of  y.  In 
deciding  upon  the  arrangement  of  signs  for  the  hyperbola  the 
student  must  remember  that  the  negative  sign  is  associated  with  1) 

and  that  ^  is  equal  to  —  i. 

i 


Analytic  Geometry.  95 

PROBLEMS. 

1.  Show  that  as  one  of  a  pair  of  conjugate  diameters  of  a 
conic  tends  to  coincide  with  one  axis  of  symmetry  the  other  tends 
to  coincide  with  the  other  axis  of  symmetry. 

2.  Show  that  if  one  of  two  conjugate  diameters  of  an  hyperbola 
meets  the  curve  in  real  points  the  other  meets  it  in  imaginary 
points,  and  conversely. 

3.  Show  that  two  conjugate  diameters  of  an  hyperbola  are  In 
the  same  quadrant,  and  find  the  angle  between  them  in  terms  of 
the  semi-axes  and  the  slope  of  one  of  the  diameters. 

4.  Make  a  similar  investigation  for  the  ellipse. 

5.  Let   two   conjugate   diameters  of  a  conic  meet  it  in  the 


points  ixi,  i/i)  and  (^x.^,  y^),  i.e.  in  the  points  (jjc^,  y^)  and  (a/—  2/i, 
xA,  and  let  the  distances  of  these  points  from  the  center 


■V 


d,'  +  d;'  =  X,'  +  y:  +  -^y,'+  "^x." 


be  denoted  by  d^  and  d^.    Then 

^     2   ,    a 
~yx  +  ^-t 

a  "^  ^ 

a        (3 

In  other  words  the  sum  of  the  squares  of  these  distances  is  con- 
stant, i.  e.,  remains  unchanged  for  all  positions  of  the  conjugate 
diameters.  In  the  ellipse  both  these  distances  are  real,  but  in  the 
hyperbola  one  or  the  other  is  imaginary.  In  consequence  of  this 
mathematicians  agree,  as  in  the  case  of  the  axes,  to  define  the 
length  of  a  semi-diameter  of  a  conic  as  the  modulus  of  the  distance 
from  the  center  of  the  conic  to  the  point  of  intersection  of  the 
diameter  and  the  conic.  It  is  usual  to  denote  the  lengths  of  a  pair 
of  conjugate  semi-diameters  by  a  and  ft'.  With  the  above  defini- 
tion in  mind,  shoAV  that  the  work  just  done  is  equivalent  to  the 
proof  of  the  theorem  : 

The  sum  I  The  difference 

of  the  squares  of  the  lengths  of  a  pair  of  conjugate  semi-diameters 

of  an 
ellipse  I  hyperbola 

is  constant  and  equal  to  the 
sum  I  difference 

of  the  squares  of  the  lengths  of  the  semi-axes. 
6.  Find  the  sines  and  cosines  of  the  angles  made  by  a  pair  of 
conjugate  diameters  with  the  axes  of  an  hyperbola,  expressing 
them  in  terms  of  a^h^  a,  h',  x^,  y^. 


96  Analytic  Geometry. 

7.  Make  a  similar  investigation  for  the  ellipse. 

8.  Show  that  the  sine  of  the  angle  between  any  pair  of  conjugate 

diameters  of  an  ellipse  is  ^, . 

ah 

9.  Make  a  similar  investigation  for  the  hyperbola. 

10.  By  aid  of  the  values  deduced  in  problems  6  and  7  find  the 
equations  of  the  ellipse  and  hyperbola  referred  to  a  pair  of  con- 
jugate diameters  as  oblique  axes  of  co-ordinates,  and  show  that  the 
equations  reduce  to  the  forms 


^-f -^=1 
a        0 


Ir-' 


69-  If  any  point  on  a  central  conic  be  joined  to 

SUPPLEMENTAL  the  extremities  of  any  diameter  two  chords 
CHORDS.  are  formed  which  are  called  supplemental 

chords.    If  the  diameter  is  the 
major  I  transverse 

axis  they  are  called  principal  supplemental  chords.  Let  (x-^,  y^) 
be  any  point  on  a  central  conic  and  {x^,  1/2)  one  of  the  extremities 
of  any  diameter.  Then  if  the  slopes  of  the  two  supplemental 
chords  determined  at  {x^,  y-^)  by  this  diameter  are  m^  and  m^ 

m,  = m,  =  — \ 


M?j  e/y 


But  the  conditions  which  must  be  satisfied  in  order  that  (x^,  y^) 
and  (x^,  2/2)  may  be  on  the  curve  give  on  subtraction 

hence  in^  m^  = ^  . 

1.  e.,  the  condition  which  must  be  satisfied  by  the  slopes  of  two 
supplemental  chords  is  the  same  which  must  be  satisfied  by  the 
slopes  of  two  conjugate  diameters.  Therefore  if  one  of  two  sup- 
plemental chords  is  parallel  to  a  diameter,  the  other  is  parallel 
to  the  conjugate  diameter. 

PROBLEMS. 

1.  Given  a  central  conic  and  a  diameter,  construct  with  ruler 
and  compass  the  conjugate  diameter,  first  from  the  definition  of 
conjugate  diameters  and  second  by  aid  of  a  pair  of  supplemental 
chords. 


Analytic  Geometry.  97 

2.  Given  a  central  conic,  find  its  center  with  ruler  and  compass. 

3.  Given  a  central  conic  and  a  diameter,  construct  by  aid  of 
this  diameter  a  pair  of  supplemental  chords  perpendicular  to  each 
other. 

4.  Given  a  central  conic  construct  its  axes  with  ruler  and 
compass. 

PROBLEMS. 

70.  1.  Following  the  analogy  of  the  work  done 

TANGENTS  AND  in  article  49,  show  that  the  condition  that 

NORMALS.  the  line 

•  y^=  mx  -\-  h 

shall  be  tangent  to  the  conic 

is         h=±\^'¥m^~+¥  I  h=±  ^/a^m^  —  'b'' 

Note  that  after  the  condition  of  tangency  has  been  imposed  on 
the  equation 

y  =  mx  +  h 

it  contains  only  one  arbitrary  parameter.  It  follows  therefore 
that  the  tangents  which  can  be  drawn  to  any  central  conic  form 
a  single  infinity  of  lines. 

2.  Following  the  method  employed  in  article  50,  develop  the 
equation  whose  roots  are  the  slopes  of  the  tangents  from  {x^,  2/1) 
to  a  central  conic.  Hence  show  that  if  the  point  is  not  on  the 
conic,  two  tangents  to  the  conic  can  be  drawn  through  it. 

3.  Following  the  method  of  the  latter  part  of  article  50,  show 
that  the  equation  of  the  tangent  to  the  conic 

ax'-V  Pif=l 

at  the  point  (a?i,  y^)  on  the  conic  is 

aa?i  X  +  Pi/i  2/  =  1 

State  the  equations  also  in  terms  of  a  and  ?>  for  the  ellipse  and 
hyperbola. 

4.  Find  the  equations  of  the  normals  at  the  same  point. 

'  5.  Show  that  the  lengths  of  the  sub-tangent  and  sub-normal  for 
the  point  {x^,  y^)  are 


^^         and  ^ 


X, 


6.  Examine  the  following  lines  for  cases  of  tangency  to 


2  _2 


-^  +  ^=1  or  ^-^=1 

9         4  9        4 

1/  +  3£P  —  4  =  0  2/  =  2 

a?_47/  +  3  =  0  5a?  — 3?/  +  3  =  0 

3y_4£P_l2==0  a?  — 3t/  +  5  =  0 


98  Analytic  Geometry. 

7.  Show  that  the  two  tangents  which  may  be  drawn  to  any 
central  conic  from  a  given  point  are  real  or  imaginarj^  according 
as  the  point  is  outside  or  inside  the  conic. 

8.  Find  the  equations  of  the  tangents  from  the  point  (3,  2) 
to  the  conies  of  problem  6. 

9.  Show  that  the  tangent  and  normal  at  any  point  bisect  the 
angles  formed  by  the  focal  radii  at  that  point. 

10.  From  the  results  of  the  last  problem  derive  a  geometric 
construction  for  the  tangent  and  normal  at  any  point  on  a  central 
conic. 

11.  Find  the  locus  of  all  points  from  which  the  two  tangents 
to  a  central  conic  are  perpendicular,  and  construct  the  locus  with 
ruler  and  compass.  Is  the  construction  always  possible  in  the  case 
of  the  hyperbola? 

12.  Find  the  locus  of  the  feet  of  the  perpendiculars  let  fall  from 
either  focus  on  the  tangents  to  an  ellipse. 

(Take  y=mx-{-h  equation  of  straight  line. 

h=  ±-\/a^m^  -\-  h^  condition  of  tangency. 

y=  —  {  —  ){x — c)  perpendicular  to  first  line 

7)1 

through  focus.     From  these  equations  eliminate  h  and  m^  and 
derive  the  desired  locus  in  the  form 


Rationalize,  replace  6  by  its  value  in  terms  of  a  and  c,  arrange 
the  equation  according  to  powers  of  c,  and  the  equation  reduces 
to 

(^2  _|_  ^2  _  ^2)  (^2  _|.  ^2  _  2ea?  +  c2)  =  0 

The  locus  is  evidently  degenerate,  representing  a  circle  concentric 
with  the  conic  and  a  pair  of  imaginary  lines  intersecting  at  the 
focus.  The  question  whether  the  real  and  imaginary  parts  of  this 
locus  correspond  to  the  perpendiculars  let  fall  upon  real  and  im- 
aginary tangents  is  left  for  the  investigation  of  the  student.) 

13.  What  changes  must  be  made  in  the  above  investigation  in 
order  to  make  it  hold  for  the  hyperbola  ? 

14.  Show  that  the  product  of  the  tAvo  perpendicular  distances 
from  the  two  foci  of  a  central  conic  to  the  tangent  at  {x^,  y^)  is 

Eliminate  y^  by  virtue  of  the  fact  that  (a?i,  i/J  is  on  the  conic, 
replace  e  by  its  value  in  terms  of  a  and  6  and  thus  reduce  the  value 
of  the  product  to  6^. 

15.  Show  that  the  tangents  at  the  extremities  of  any  diameter 
are  parallel  to  the  conjugate  diameter. 


Analytic  Geometry.  99 

16.  By  aid  of  the  last  problem  and  problem  8,  article  68,  show 
that  the  area  of  the  parallelogram  formed  by  the  tangents  at  the 
extremities  of  any  pair  of  conjugate  diameters  of  a  central  conic 
is  independent  of  the  position  of  the  diameters  and  equal  to  the 
area  of  the  rectangle  formed  by  the  tangents  at  the  extremities  of 
the  axes. 

71.  Follow  the  methods  used  in  the  investiga- 

POLES  ANDPOLARS.  tion  of  poles  and  polars  of  the  circle,  and 
solve  the  following: 

PROBLEMS. 

1.  Find  the  equation  of  the  polar  of  (a?i,  y^)  with  respect  to  the 
conic 

ajf  +  %"  =  1. 

2.  Find  the  co-ordinates  of  the  pole  of 

aw  -^1)1/  -\-  c  =  0 
with  respect  to  the  conic 

3.  Show  that  if  the  pole  of  a  given  line  with  respect  to  a  given 
central  conic  is  on  the  conic  the  polar  is  the  tangent  at  the  pole. 

4.  Show  that  if  a  point  lies  on  its  own  polar  with  respect  to  a 
central  conic,  it  lies  also  on  the  conic. 

5.  Show  that  if  the  polar  of  a  point  A  passes  through  B  the 
polar  of  B  passes  through  A. 

6.  Given  any  pair  of  conjugate  diameters  of  a  central  conic  show 
that  the  polars  of  all  points  on  the  one  are  parallel  to  the  other. 

7.  Give  geometric  constructions  for  poles  and  polars  in  the  case 
of  both  ellipse  and  hyperbola. 

8.  Show  that  the  directrices  are  the  polars  of  the  foci. 

9.  If  it  can  be  shown  that  any  investigation  does  not  depend 
on  the  rectangularity  of  the  axes,  the  results  of  the  investigation 
hold  good  for  oblique  axes.  Show  in  this  way  that  the  form  of  the 
equation  of  the  polar  developed  above  holds  good  so  long  as  the 
central  conic  is  referred  to  a  pair  of  conjugate  diameters  as  co- 
ordinate axes. 

72.  If  in  the  equation  of  the  tangent  at  any 

ASYMPTOTES.  point  (a?i,  i/i)  on  a  central  conic  we  substitute 

the  value  of  p^,  deduced  from  the  fact  that 

(^1?  Vi)   is  a  point  on  the  conic,  we  have  a  furm  which   reduces 

at  once  to 


ax±  ij 


100  Analytic  Geometry. 

The  double  sign  before  the  radical  arises  from  the  fact  that  there 
are  two  values  of  i/i,  and  hence  two  tangents,  corresponding  to  a 
single  value  of  x^.  If  now  the  value  of  a?i  is  allowed  to  increase 
indefinitely,  i.  e.,  if  the  point  of  tangency  is  allowed  to  recede 
indefinitely  from  the  origin,  this  equation  of  the  tangent  tends  to 
the  limiting  form 

or  l)x  ±  iay  =  0  I  hx  ±:ay=^0 

In  other  words  the  tangents  at  infinity  to  the  ellipse  are  a  pair 
of  imaginary  lines  intersecting  in  the  center  of  the  ellipse,  and  for 
the  hyperbola  a  pair  of  real  lines  intersecting  in  the  center  of  the 
hyperbola  and  forming  the  diagonals  of  the  rectangle  on  the  axes. 
Tangents  at  infinity  to  any  central  conic  are  called  its  asymptotes.* 

PKOBLEMS. 

1.  Find  the  angle  between  the  two  asymptotes  of  a  central  conic. 

2.  Show  that  the  asymptotes  of  an  equilateral  hyperbola  are  per- 
pendicular to  each  other.  From  this  fact  the  equilateral  hyper- 
bola is  sometimes  called  a  rectangular  hyperbola. 

3.  Show  that  any  asymptote  regarded  as  a  diameter  is  its  own 
conjugate. 

4.  Show  that  any  two  conjugate  diameters  of  an  hyperbola  are 
separated  by  an  asymptote. 

5.  Show  that  if  the  asymptotes  of  an  hyperbola  be  taken  as  a 
pair  of  oblique  axes  the  equation  of  the  hyperbola  reduces  to  the 
form 


__«'  +  J) 


rj^y QJ.  g,y    = 

according  to  the  choice  of  positive  directions  of  the  new  axes. 

6.  Show  that  the  equation  of  the  tangent  at  any  point  (a?i,  y^) 
on  the  hyperbola  has  the  form 


*This  definition  might  be  made  a  general  definition  of  an  asymptote  to 
any  curve  were  it  not  for  the  fact  that  in  special  cases  the  line  at  infinity 
is  itself  a  tangent,  and  some  mathematicians  prefer  to  exclude  it  from  the 
list  of  possible  asymptotes. 


Analytic  Geometry.  101 

^x  Vx 

if  the  asymptotes  are  the  axes  of  co-ordinates.* 

7.  Show  that  the  segment  of  the  tangent  included  between  the 
asymptotes  is  bisected  at  the  point  of  tangency.f 

8.  Show  that  the  product  of  the  intercepts  of  the  tangent  on  the 
asymptotes  is  independent  of  the  position  of  the  tangent  and  equal 
to  the  sum  of  the  squares  of  the  lengths  of  the  semi-axes. 

9.  Show  that  the  area  of  the  triangle  formed  by  the  asymptotes 
and  any  tangent  is  independent  of  the  position  of  the  tangent  and 
equal  to  the  product  of  the  lengths  of  the  semi-axes. 

73-  Closely  associated  with  the  hyperbola 

THE  CONJUGATE 

HYPERBOLA.  ^  _  ^  ::=  i 

2      72-^ 

a       0 
there  is  a  second  hyperbola 

2  2 

^^    J-    r=1 

2    ^^     7  2  -*- 

a        0 

which  is  turned  along  the  Y  axis  in  place  of  the  X  axis  and  has  the 
transverse  and  conjugate  axes  of  the  original  hyperbola  as  con- 
jugate and  transverse  axes.  It  is  called  the  conjugate  of  the 
original  hyperbola  and  plays  an  interesting  part  in  the  theory  of 
conjugate  diameters. 

PKOBLEMS. 

1.  Show  that  if  a  diameter  meets  an  hyperbola  in  real  points 
its  conjugate  meets  the  conjugate  hyperbola  in  real  points  and 
conversely. 

2.  Show  that  an  hyperbola  and  its  conjugate  have  the  same 
asymptotes. 


*The  student  will  find  it  simpler  to  deduce  the  equation  of  the  tangent 
directly  from  the  equation 

than  to  deduce  it  by  transformation  of 

t Remember  that  while  the  formula  for  distance  presupposes  rectangularity 
the  formulae  for  division  of  a  segment  in  a  given  ratio  do  not. 


102 


Analytic  Geometry. 


3.  Show  that  if  a  diameter  meets  an  hyperbola  in  imaginary 
points  it  meets  the  conjugate  hyperbola  in  real  points  whose  co- 
ordinates are  the  moduli  of  the  co-ordinates  of  the  imaginary 
points  in  which  it  meets  the  original  hyperbola. 

4.  Show  that  the  asymptotes  separate  the  diameters  of  an 
hyperbola  into  tw^o  groups  (whose  slopes  are  respectively  less  and 

greater  than— ) one  meeting  the  original  hyperbola  in  real  points 

a 
and  the  other  meeting  the  conjugate  hyperbola  in  real  points. 

74.  The  two  circles  having  the  center  of  the 

THE  AUXILIARY       '  ellipse  as  centers  and  the  major  and  minor 
CIRCLES.  axes  of  the  ellipse  as  diameters  are  called  the 

major  and  minor  auxiliary  circles.  The 
practical  importance  of  these  circles  in  the  theory  of  the 
ellipse  is  largely  due  to  the  fact  that  they  serve  to  connect 
the  ellipse  with  the  circle,  a  curve  for  which  w^e  have  a  full 
and  for  the  most  part  simple  geometric  treatment.  The  corre- 
sponding curves  for  the  hyperbola  are  a  pair  of  equilateral  hyper- 
bolas, and  the  lack  of  a  corresponding  geometric  treatment  for 
equilateral  hyperbolas  renders  the  analogues  of  the  following  prob- 
lems of  small  practical  importance : 


PEOBLEMS. 

1.  Show  that  if  a  point  on  the  ellipse  and  one  on  the  major 
auxiliary  circle  have  equal  abscissas  their  ordinates  are  in  the 

ratio — . 
a 

2.  Given  an  ellipse  and  its  major 
auxiliary  circle,  divide  the  major 
axis  into  a  number  of  equal  parts 
and  on  these  parts  construct  pairs 
of  rectangles  of  which  AD  and  AF 
are  a  sample  pair.  From  the  last 
problem  the  areas  of  any  such  pair 

are  evidently  in  the  ratio  —  .   But 

a 
the  areas  of  the  ellipse  and  the 
major  auxiliary  circle  are  evidently 
twice  the  limits  of  the  sums  of  these 
rectangles  as  the  number  of  parts 
into  which  the  major  axis  is  divided 
tends  to  infinity.     Show  from  this  fig.  29. 

that  the  area  of  the    ellipse  is  irah. 


Analytic  Geometry.  lOfi 

3.  Find  the  areas  of  the  ellipses  of  problems  2  and  3,  article 
67. 

4.  If  through  any  point  D  on  an  ellipse  the  ordinate  be  drawn 
and  extended  till  it  meets  the  major  auxiliary  circle  in  the  point 
F,  then  the  angle  FOX  is  called  the  eccentric  angle  of  the  point  D. 
Show  that  if  {x^,  y-^)  is  any  point  on  an  ellipse  and  ^^the  eccentric 
angle  of  that  point 

Xy=^a  cos 0^  2/i  =  ^  si^ 0x 

5.  In  our  investigation  of  the  sub-tangent  it  was  found  that 
the  length  of  the  sub-tangent  corresponding  to  the  point  [x^,  i/j 
did  not  depend  on  either  5  or  y^.  It  follows  therefore  that  if  a 
family  of  ellipses  is  constructed  with  the  same  major  axis  but 
with  different  minor  axes,  an  ordinate  erected  at  any  point  in 
the  major  axis  will  cut  this  family  of  ellipses  in  a  series  of  points 
having  the  same  sub-tangents,  i.  e.,  the  tangents  at  all  these  points 
meet  at  the  same  point  on  the  major  axis.  This  family  of  ellipses 
includes  the  major  auxiliary  circle.  Hence  derive  a  method  of 
drawing  a  tangent  at  any  point  on  an  ellipse  with  ruler  and 
compass. 


CHAPTER  XV. 


75. 
DETERMINATION 
OF  FORM. 


THE  PARABOLA. 

We  take  up  now  the  consideration  of  type 
(2)  of  Chapter  XIII,  the  equation 

By'+2Gx  =  0, 

n 
in  which  neither  B  nor  G  is  zero.    Solve  fovy,  put —  =c,  and 

we  have 

y^=  2cx, 

If  c  is  positive  the  values  of  y  are  imaginary  when  x  is  negative, 

zero  when  x  is  zero,  and  real  when  x  is  positive.    As  x  increases 

from   zero    to    plus    infinity    y   also 

increases   to   plus    infinity    for    one 

set  of  values  and  decreases  tc  minus 

infinity  for  the  other  set.     The  curve 

therefore  lies  wholly  to  the  right  of 

the  Y  axis,  passes  through  the  origin 

and  extends  upward  and  downward 

ind-efinitely.      Careful    plotting   will 

show  it  to  be  of  the  form  here  given. 

It    is    called    a    parabola.     If    c    is 

negative,    all    that    has    been    said 

holds  except  that  the  curve  lies  to  the 

left  of  the  Y  axis. 


Fig.  30. 


76.  The  parabola,  like  the  ellipse  and  the  hy- 

EARLY  GEOMETRIC    perbola,  was  known  to  geometricians  before 
DEFINITION.  the  discovery  of  analytic  geometry.     One  of 

its  geometric  definitions  is  as  follows:  The 
parabola  is  the  locus  of  a  point  moving  in  such  a  way  that 
its  distance  from  a  fixed  point,  divided  by  its  distance  from  a 
fixed  line,  is  equal  to  unity.  As  in  the  case  of  the  ellipse  and 
hyperbola  the  fixed  point  is  called  the  focus,  the  fixed  line  the 


Analytic  Geometry. 


105 


directrix  and  the  constant  ratio  the  eccentricity.  Selecting  for 
axes  of  co-ordinates  the  directrix  and  the  perpendicular  upon  it 
from  the  focus,  and  denoting  the  distance  from  the  focus  to  the 
directrix  by  d^  we  deduce  at  once  from  the  definition  just  given 
the  equation 

y^  =  2d{Q0-r  ^  ). 

The  form  of  this  equation  shows  at  once  that  the  X  axis  is  an  axis 
of  symmetry,  that  the  Y  axis  is  parallel  to  the  other  axis  of  sym- 
metry, and  that  the  vertex  of  the  curve  is  at  the  point  ( — ,  0) . 

Moving  the  Y  axis  parallel  to  itself  till  it  passes  through  this 
vertex  we  reduce  the  equation  to  the  form 

y^  =  2dx 
which  corresponds  to  the  form  of  the  previous  article.  The  defini- 
tion of  the  curve  given  above  shows  also  that  the  distance  d  equals 
the  semi-parameter  p,  as  defined  in  article  67,  problem  9.  We 
have  therefore  the  following  relations  between  the  constants  we 
have  employed :  c^^d=p. 

By  means  either  of  the  geometric  definition  of  the  parabola  or 
by  consideration  of  the  work  done  in  problems  11,  12,  13,  article 
67,  we  see  at  once  that  the  parabola  is  the  limiting  form  of  either 
the  ellipse  or  hyperbola  as  the  center  tends  to  infinity  under  the 
restriction  p  =  constant,  or  as  the  eccentricity  tends  to  unity. 
The  student  will  find  it  both  interesting  and  profitable  to  derive 
theorems  for  the  parabola  by  an  examination  of  the  limiting  form 
of  the  corresponding  theorems  for  the  ellipse  or  hyperbola. 


77. 
MECHANICAL 
CONSTRUCTION. 


To  trace  the 
parabola  mechan- 
ically we  place 
one  edge  of  a  rect- 
angular board  CE 
against  the  directrix  AB^  fasten  one 
end  of  a  cord  equal  in  length  to  CD  at 
D  and  the  other  end  at  the  focus  F^  and 
slide  the  board  along  the  directrix,  keep- 
ing the  cord  pressed  against  the  edge 
CD  by  a  pencil  point  P.  The  point  P 
will  trace  the  parabola,  for  in  every 
position  we  have  PF  =  PC. 


B 

T, 

.  C 

^.^-^ 

d 

(/ 

A 

\r 

Fig.  31. 


106  Analytic  Geometry. 

PKOBLEMS. 

1.  Show  that  the  vertex  of  a  parabola  is  midway  between  the 
focus  and  the  directrix. 

2.  Find  the  focal  radius  of  any  point  (a?i,  y-^)  on  the  parabola 

3.  Find  the  foci,  parameters,  and  directrices  of  the  following 
parabolas. 

y'^=l{)x  y^  = — 4a? 

y'^=ax  -{-1)  ^^  =  — 4^+5 

x^  =  4:y  a)^=3y-{-l 

y^  —  x  +  l  =  0  a?2  +  22/  — 1  =  0 

1/2  =  3ip  +  4  Sy^-\-4.x—2  =  0 

4.  Write  the  equations  of  the  following  parabolas : 

X  —  2  =  0  is  the  directrix,  and  (3,  4)  the  focus. 

(3,  4)  is  the  focus,  axis  of  symmetry  is  parallel  to  the  Y 

axis,  curve  opens  upward,  parameter  is  6. 
(2,  3)  is  the  vertex,  (1,  3)  the  focus. 

78.  The  equation  of  the  diameter  of  the  para- 

DIAMETERS.  bola  may  be  regarded  as  a  special  case  of  the 

general  equation  developed  in  article  62,  or 
it  may  be  found  directly  by  determining  the  locus  of  the  middle 
points  of  a  system  of  parallel  chords.  If  the  latter  method  is 
adopted,  the  student  should  remember  that  in  any  locus  problem 
the  thing  sought  is  the  equation  which  determines  the  movement 
of  the  tracing  point.  If  then,  as  in  the  present  instance,  there  is 
developed  in  the  course  of  the  discussion  an  equation  of  the  type 

y=1c 

(where  k  is  a  constant  and  i/  is  a  co-ordinate  of  the  tracing  point) 
this  equation  in  itself  determines  the  path  of  the  tracing  point, 
and  is  therefore  the  equation  desired. 

PROBLEMS. 

1.  Write  the  equation  of  the  diameter  of  the  parabola  y^==2px 
bisecting  the  chords  of  slope  m. 

2.  Given  a  parabola  y^  =  2px^  and  a  diameter  y^=^a,  what  is 
the  slope  of  the  chords  which  it  bisects? 

3.  Show  that  all  diameters  of  a  parabola  are  parallel  to  the  axis 
of  symmetry ;  first  from  the  form  of  the  equation  of  the  diameter, 
and  then  from  the  location  of  the  center. 

4.  Why  do  we  not  develop  for  the  parabola  the  theory  of  con- 
jugate diameters  and  supplemental  chords? 


Analytic  Geometry.  107 

79-  PKOBLEMS. 

TANGENTS  AND 

NORMALS.  1-  What  condition  must  be  satisfied  by  the 

coefficients  of  the  line 

in  order  that  it  may  be  tangent  to  the  parabola 

2.  Develop  the  equation  whose  roots  are  the  slopes  of  the  tan- 
gents from  the  point  {x^,  y^)  to  the  parabola 

2/2  =  2px 

Hence  show  that  if  the  point  is  not  on  the  conic  two  tangents  to 
the  conic  can  be  drawn  through  it. 

3.  Show  that  the  equation  of  the  tangent  to  the  parabola 

1/2  =-  2px 
at  the  point  (^i,  i/J  is 

4.  Find  the  equation  of  the  normal  at  the  same  point. 

5.  Find  the  length  of  the  sub-tangent  and  show  that  it  is 
bisected  at  the  vertex. 

6.  Show  that  the  subnormal  is  constant  and  equal  to  the  semi- 
parameter. 

7.  Hence  deduce  geometric  constructions  for  the  tangent  and 
normal  at  any  point  of  a  parabola  whose  focus  and  axis  are  given. 

8.  Show  that  the  two  tangents  which  may  be  drawn  from  any 
point  to  a  parabola  are  real  or  imaginary  according  as  the  point 
is  without  or  within  the  parabola. 

9.  Show  that  the  tangent  and  normal  at  any  point  bisect  the 
angles  formed  by  the  focal  radius  of  the  point  and  the  diameter 
through  the  point.  Show  also  that  this  is  a  special  case  of  problem 
9,  article  70. 

10.  Hence  deduce  geometric  constructions  for  the  tangent  and 
normal  at  any  point  on  a  parabola  whose  focus  and  axis  are  given. 

11.  Note  that  problem  9  is  equivalent  to  the  statement  that  the 
tangent  makes  equal  angles  with  the  axis  of  symmetry  and  the 
focal  radius  through  the  point  of  tangency,  and  hence  deduce  a 
geometric  construction  for  the  tangent  and  normal. 

12.  Find  the  locus  of  the  points  from  which  two  perpenliicular 
tangents  may  be  drawn  to  a  parabola. 

13.  Find  the  locus  of  the  foot  of  the  perpendicular  let  fall  from 
the  focus  upon  a  variable  tangent  to  the  parabola,  and  show  that  it 
is  degenerate  and  consists  of  the  tangent  at  the  vertex  and  a  pair 
of  imaginary  lines. 


108  Analytic  Geometry. 

14.  Hence  deduce  a  geometric  construction  for  the  tangent  at 
any  point  of  a  parabola  whose  focus  and  axis  are  given. 

15.  ShoAV  that  problems  12  and  13  are  special  cases  of  the  results 
obtained  in  problems  11  and  12  of  paragraph  70. 

16.  Show  that  the  perpendicular  distance  from  the  focus  to  any 

tangent  is  equal  to-z^\/2p7-,  where  r  is  the  focal  radius  of  the  point 

of  tangency. 

17.  Show  that  the  tangent  at  the  extremity  of  any  diameter  is 
parallel  to  the  chords  bisected  by  that  diameter. 

18.  Deduce  the  equation  of  the  parabola  referred  to  any  diam- 
eter and  the  tangent  at  the  extremity  of  that  diameter  as 
oblique  axes,  and  show  that  it  reduces  to  the  form 

y^  =  2px, 

where  p  is  a  new  constant.    What  is  the  value  of  p  in  terms  of  p 
and  the  angle  between  the  new  axes  ? 

19.  The  slope  of  the  tangent  to  the  parabola  y-  =  2px  at  the 

point  (a?],  2/i)  is  —  .    Since  (x-^,  y^)  is  on  the  conic  this  may  be  re- 


y 


duced  to   \P_  ^    Show  that  when  the  point  of  tangency  removes 

toward  infinity  the  tangent  tends  to  parallelism  with  the  axis  of 
symmetry  of  the  parabola. 

so.  Build  up  the  theory  of  poles  and  polars  for 

POLES  AND  POLARS.  the  parabola. 


CHAPTER  XVI. 

ADDITIONAL  WOEK  ON  THE  GENERAL  EQUATION  OF 
THE  SECOND  DEGREE. 

81-  In   Chapter   XIII   we    investigated   such 

NECESSITY  OF  A      properties  of  the  curve  represented  by  the 
GENERAL  general  equation   of   the   second    degree  as 

TREATMENT.  would  guide  US  in  reducing  the  equation  to 

certain  standard  or  type  forms.  By  the  aid 
of  these  type  forms  any  investigation  which  deals  with  the  curve 
as  an  individual  and  without  regard  to  its  relative  position  may 
evidently  be  carried  on.  For  example,  the  theorem  that  the  locus 
of  the  points  of  intersection  of  perpendicular  tangents  to  an  ellipse 
is  a  circle  concentric  with  the  ellipse  is  altogether  independent 
of  the  position  of  the  ellipse,  and  therefore  is  proved  with  entire 
generality  by  the  use  of  the  type  form.  On  the  other  hand,  any 
theorem  which  depends  upon  the  position  of  a  conic  with  respect 
to  the  axes  or  to  other  curves  cannot  be  established  in  its  most 
general  form  by  the  use  of  type  forms,  since  these  presuppose  a 
special  position  of  the  curve  with  respect  to  the  axes.  For 
example,  the  condition  of  tangency,  equation  of  polar,  values  of 
sub-tangent  and  sub-normal  hitherto  deduced  are  all  based  upon 
the  assumption  that  the  curve  occupies  a  particular  position,  and 
therefore  they  cannot  be  applied  to  the  same  curve  in  any  other 
position.  So  long  as  we  are  dealing  with  a  single  curve  there  is 
no  reason  why  we  should  not  assume  a  particular  position  for  it, 
but  when  two  or  more  curves  enter  into  the  discussion  only  one 
of  them  in  general  can  be  given  the  desired  position,  and  it  is 
therefore  necessary  to  make  an  investigation  of  the  conic  without 
regard  to  its  position,  i.  e.,  to  make  a  study  of  the  general  equation 
of  the  second  degree. 

The  results  already  secured  are  contained  in  a  series  of  theorems 
in  the  opening  paragraphs  of  Chapter  XIII. 


110  Analytic  Geometry. 

82.  When  the  left  hand  member  of  an  equation 

DEGENERATE  consists  of  two  OF  more  rational  factors,  the 

CONICS.  equation  is  satisfied  by  any  set  of  values  of 

the  variables  which  make  any  one  of  the 
factors  equal  to  zero.  In  consequence  the  locus  corresponding  to 
such  an  equation  consists  of  two  or  more  parts,  which  are  them- 
selves the  loci  representing  the  equations  formed  by  equating  the 
various  factors  to  zero.  Such  a  locus  is  said  to  be  degenerate. 
When  a  conic  degenerates  it  is  evident  that  the  factors  of  the  left 
hand  member  must  be  of  the  first  degree  and  hence  that  the  conic 
must  degenerate  into  a  pair  of  straight  lines.  The  only  questions 
of  interest  in  such  a  case  are  concerning  the  point  of  intersection 
and  the  angle  between  the  lines.  The  point  of  intersection 
evidently  meets  the  definition  of  the  center  of  symmetry  and 
consequently  may  be  found  by  solving  the  equations 

Jix  +  Z>i/  +  f  =  0 

To  find  the  angle  between  the  lines  assume  that  the  conic  degener 
ates  into  the  two  lines 

a^x  +  ^i//  +  Yi  =  0 
a,x  +  ^2//  +  72  =  0. 

Then  the  equation  of  the  conic  is 

+  (a^y,  +  a,y,},io  +  (fty,  +  fty,)//  +  Tiy.,  =  0 

Consider  the  two  lines  parallel  to  these  lines  and  passing  through 
the  origin 

a,x  +  A?/  =  0 
a.^x  +  /3,f/  =  0 

These  taken  together  constitute  a  degenerate  conic 

a,a.y  +  (3,^,!/'  +  (a,/?,  +  a^^ri/  =  0 

We  may  at  once  state  the  theorem  that  the  second  degree  part  of 
the  equation  of  a  degenerate  conic  is  the  left  hand  member  of  the 
equation  of  a  second  degenerate  conic  whose  lines  intersect  in  the 
origin  and  are  parallel  to  the  lines  of  the  first  conic. 


Analytic  Geometry.  Ill 

It  follows  that  if 

0^2  +  &  ?/2  4-  2hxy  -\-2gx-{-2fy-\-c  =  ^ 

represents  a  degenerate  conic,  the  angle  between  the  lines  is  equal 
to  the  angle  between  the  lines  of 

ax''  +  lif  +  2}ixy  =  0 

This  equation  may  be  at  once  factored  and  the  angle  0  between 
the  lines  expressed  by  the  relation 


a  -\-  0 

83.  It  is  desirable  to  have  some  test  to  apply 

DISCRIMINANT.  to  a  second  degree  equation  in  order  to  deter- 

.  mine  at  once  whether  or  not  it  is  degenerate. 
A  moment's  consideration  will  show  that  one  peculiarity  of  a 
degenerate  conic  is  that  the  center  of  symmetry,  i.  e.,  the  point  of 
intersection  of  the  two  component  lines,  is  on  the  conic;  and, 
conversely,  the  center  of  symmetry  can  be  on  the  conic  only  when 
the  conic  degenerates.  The  necessary  and  sufficient  condition  for 
degeneracy  of  a  conic  is  therefore  the  existence  of  a  point  which 
satisfies  the  three  equations 

r  ax  +  hy^g  =  ^ 

[  ax^  +  hy^  +  2}ixy  +  2^0?  +  2fi/  +  c  =  0. 
The  last  of  these  may  be  written  in  the  form 

x{ax-\-hy-{-g)-{-y(hx^-hy^t)^gx-^fy-\-c=:Q 
Equations  I  are  therefore  equivalent  to 

ax  +  hy  -\-g  =  0 

9^  +  fy+c  =  0. 
But  the  necessary  and  sufficient  condition  for  the  simultaneous 
satisfaction  of  these  three  equations  is 

a        h        g 

h        h        f   =0. 

9        f        c 

This  determinant,  which  plays  an  important  part  in  the  theory  of 
conies,  is  called  the  discriminant  of  the  equation  of  the  conic. 
We  shall  denote  it  by  A. 


112  Analytic  Geometry. 

84.  After  it  has  been  shown  that  a  conic  is  not 
CLASSIFICATION.        degenerate,  the  most  important  question  is 

whether  it  is  an  ellipse,  hyperbola,  or  para- 
bola. Article  72  and  Problem  19,  article  79,  have  shown  that  the 
ellipse  has  only  imaginary  points  at  infinity,  that  the  hyperbola 
extends  to  infinity  in  two  different  directions,  and  that  the  two 
sides  of  the  parabola  tend  to  parallelism  as  the  tracing  point  goes 
off  to  infinity.  If  then  we  join  the  infinite  points  on  a  conic  to 
any  point  in  the  plane  the  resulting  pair  of  lines  will  have 
imaginary  slopes  for  the  ellipse,  real  and  equal  slopes  for  the 
parabola,  real  and  unequal  slopes  for  the  hyperbola.  The  lines 
which  run  from  any  point  to  the  infinite  points  on  the  conic  are 
the  lines  which  give  infinite  values  for  r  in  the  equation  of  article 
59,  i.  e.,  the  lines  whose  slopes  are  such  that  we  have 

al''  +  21ilm  +  l)m^  =  {). 

The  factors  of  this  are  imaginary,  real  and  equal,  or  real  and  un- 
equal according  as  ah  —  Ji^  is  positive,  zero,  or  negative.  An  equa- 
tion of  the  second  degree  in  two  variables  therefore  represents  an 
ellipse,  parabola,  or  hyperbola  according  as  ab  —  Ji^  is  positive, 
zero,  or  negative. 

85.  Since  the  tangent  is  a  line  meeting  the 
TANGENTS  AND          conic  in  two  Coincident  points,  the  condition 

that  a  line  through  {x^,  y^)  may  be  tangent 
to  the  conic  represented  by  the  general  equa- 
tion of  the  second  degree  is  that  the  two   values  of  r  given  by 
the  equation 

t(^i,  Vi)  +  2r[l{ax,  +  hij,  +  g)+  m{hx,  +  l)y^  +  f) ] 
+  r^aP  +  2hlm  +  hm^)=0* 

shall  be  equal.    The  condition  for  equal  roots  gives 

I  fi^i,  Vi)  («^'  +  2/jim  +  6m2) 

—[l{a(D^  +  hy^  +  g)  +  m(hx^+  &t/i  +  f)]2  =  0, 

an  equation  of  the  second  degree  in  the  ratio  —  whose  two  roots 

are  the  slopes  of  the  two  tangents  from   {x^,  y^)   to  the  conic. 


'f(^uyi)  =  a'^i  +  W-^^^^iyi  +  ^93: 1  +2/2/1  +  c. 


Analytic  Geometry.  113 

Eliminating  ^  between  this  equation  and  the  equation  of  the  line 
through  (a?i,  2/1) 

II  x  —  x,  ^  y  —  y, 

I  771 

we  have 

III  /(a?,,  y^)[a{x  —  x^y-\-2h{x—x^)(y  —  y^)-\-'b{y  —  y^y] 
—  [(^  —  ^i)(«^i  +  %i  +  5')  +  (2/  — 2/1)  (^^^1  +  ^2/1 +  f)]'=0 

as  the  equation  of  the  pair  of  tangents  from  {x^,  y^)  to  the  conic. ^ 
If  (^ij  Vi)  is  on  the  curve,  f(x^,  y^)=0^  the  two  tangents  coin- 
cide, the  equation  of  the  pair  of  tangents  reduces  to  a  perfect 
square,  and  we  have  as  the  equation  of  the  tangent  at  the  point 
(a?i,  1/1)  on  the  conic 

{X  —  X,)  {ax^  +  hy^  +  9)  +  {y—Vx)  (^^1  +  &2/i  +  f )  =  0. 

By  transposing  the  negative  terms  to  the  right  hand  member  and 
adding 

gxx  +  fyx  +  G 

to  both  sides,  this  last  equation  is  reduced  to  the  form 

xiax^  +  hy^  +  g)^y{hx^  +  hy^  +  f)  +  g3o^  +  fy^+c  = 
x^  ( ax^  -\-hy^  +  g)  -\-  y^  ( hx^  +  l)y^  +  t)  +  gx^  -{.fy^-\-G  = 

we  have  therefore  for  the  equation  of  the  tangent  at  the  point 
(a?i,  2/1)  on  the  curve 

x{ax^  +  %i  +  g)  +  y{hx^  +  fti/i  +  /)  +  g^i  +  f^i  +  c  =  0, 

which  may  also  be  written  in  the  form 

x^{ax  +  hy  +  g)-\-  y^ihx  +  hy  -^  f)-{- gx  -]-fy  +  c  =  0,       . 

as  the  student  will  see  on  multiplying  out  the  two  forms. 

The  normal  at  any  point  on  the  conic  is  the  perpendicular  to 
the  tangent  at  that  point.  With  this  definition  the  student  should 
have  no  difficulty  in  writing  its  equation. 


*The  significance  of  this  elimination  may  not  seem  clear  to  the  student. 
Equation  II  states  that  the  point  {x,  y)  is  on  a  line  through  {x^,  y^)  with 
direction  cosines  I,  m.  Equation  I  states  that  I  and  m  are  so  determined 
that  the  line  is  tangent.  Equation  III,  deduced  by  making  I  and  II  simul- 
taneous,  states   that    [x,   y)    is   on   one  or  the  other  of  the  tangent  lines. 


114 


Analytic  Geometry. 


86. 
A    SECOND 
CONDITION   OF 
TANGENCY. 

first  degree.    Let 


The  condition  of  tangencv  given  in  the  last 
article  is  applicable  when  we  have  a  line 
through  a  fixed  point.  We  frequently  need 
the  condition  of  tangencv  applicable  when 
the  line  is  given  hj  a  general  equation  of  the 

ax-i-  I3y  +  1  =  0 


be  a  given  straight  line,  what  is  the  condition  which  must  be  sat- 
isfied in  order  that  it  may  be  tangent  to  the  conic 

ax^  +  ly^  +  2}ixy  -\-2gx +  2fy  +  c  =  01 

Let  us  assume  the  line  to  be  tangent  at  a  point  (x^,  y^).    Its 
equation  must  then  be 

x(ax^  +  hy^  +  g)  +  yUix^  +  hy^  +  /)  +  gx^  +  /t/^  +  c  =  0. 

This  equation  represents  the  same  line  as 

ax  +  I3ij  +1  =  0 
Therefore 

ax^  +  A?/i  +  f/  __  hx^  +  hpi  +  f f/Xi  +  fy^  +  c 

a  ^  ~  1 

For  convenience  of  elimination  equate  each  of  these  ratios  to  —  /x 
and  we  have 

'^^i  +  ft^i  +  f  +  ^/^^O 

a.^1  +  ^Vi  +1  =  0 

the  last  of  which  holds  true  because  the  point  {x^^  y^)  is  on  the 
line 

ax  +  ^y  +  1  =  0. 

The  necessary  and  suflScient  condition  for  the  co-existence  of  these 
four  equations  is 

a     h     g    a\ 

li     h     f    ^ 

g     f    c     1 

a    i8    1     0 


=  0 


which  is  therefore  the  condition  which  must  be  satisfied  by  the  co- 
efficients of  the  given  line  and  conic  in  order  that  they  may  be 
tangent  to  each  other. 


Analytic  Geometry.  115 

Expand  this  determinant,  arrange  the  terms  according  to  powers 
of  a  and  ^,  denote  the  minors  of  the  determinant,  taken  with 
their  proper  signs,  by  A,  B,  C,  F,  G,  H,  as  usual,  and  the  condition 
of  tangency  takes  the  form 

Aa'  +  B(i'  +  2Hali  +  2(9a  +  2F^  +  6^=  0  * 
a  form  whicli  we  shall  hereafter  denote  by 

From  this  general  condition  of  tangency  the  student  may  at  once 
deduce  as  special  cases  the  conditions  developed  in  articles  49,  70, 
and  79. 

What  is  the  vital  difference  between  the  equation  just  developed 
and  the  equation  of  the  conic?  f{x,  y)^0  is  a  condition  which 
must  be  satisfied  by  the  co-ordinates  x  and  y  of  all  points  on  the 
conic,  i.  e.,  the  condition  which  selects  from  all  the  points  in  the 
plane  those  that  lie  on  the  conic.  Similarly,  F(ia,  /?)  =  0  is  the 
condition  which  must  be  satisfied  by  the  co-efficients  of  every  line 
tangent  to  the  conic,  i.  e.,  the  condition  which  selects  from  all 
the  lines  in  the  plane  those  which  are  tangent  to  the  conic.  But 
a  conic  is  just  as  fully  determined  by  the  aggregate  of  its  tangents 
as  by  the  aggregate  of  its  points.  The  following  questions  im- 
mediately present  themselves,  (a)  Since  the  co-efficients  a  and 
13  determine  the  position  of  the  line,  are  they  not  in  some  sense 
co-ordinates  of  the  line?  (b)  Is  not  F(a,  /S)  =  0  just  as  truly 
the  equation  of  the  conic  as  /"(^^  i/)  =  0?  (c)  Is  it  not  possible  to 
build  up  a  geometry  in  which  the  variable  element  is  the  line 
rather  than  the  point?  (d)  If  so,  would  not  the  algebraic  work 
of  the  two  geometries  be  identical;  and,  therefore,  could  we  not 
infer  from  each  theorem  already  developed  a  new  one  differing 
from  the  old  by  an  interchange  of  point  and  line?  The  answers 
to  these  questions  lie  beyond  the  scope  of  the  present  volume,  but 
the  student  Avho  will  follow  them  up  w^ill  find  that  they  lead  into 
one  of  the  most  interesting  fields  of  modern  mathematical  investi- 
gation. 


*An  immediate  expansion  of  the  determinant  in  this  form  is  possible. 
Consult  in  any  standard  text  on  determinants  the  theorem  on  the  expansion 
of  a  determinant  in  terms  of  the  products  in  pairs  of  tlie  constituents  of 
any  row  and  column. 


116  Analytic  Geometry. 

87.  Any  one  of  the  methods  used  for  deter- 

POLE  AND  POLAR,  mining  the  equation  of  the  polar  of  a  point 
with  respect  to  a  conic  might  be  extended 
to  the  general  conic.  We  consider  the  one  which  regards  the  polar 
as  the  locus  of  harmonic  conjugates  of  the  pole  with  respect  to  the 
intersections  of  the  conic  and  the  chords  through  the  pole.  (This 
method  was  developed  for  the  circle  in  article  55,  and  the  notation 
and  diagram  of  that  article  will  serve  for  this  also.)     Let 

ax''  +  &I/2  +  21ixy  +  2gx +  2fij +  c  =  Q 

be  any  conic,  and  {x^,  t/J  any  point  P,  Write  the  equations  of 
any  line  through  {x-^,  y-^)  in  the  form 

y  =  y^-\-mr. 

Then  the  distances  from  the  point  {x^,  y^)  to  the  conic,  measured 
along  this  line,  are  given  by  the  equation 

/(^i^  yx)+2r[l{ax^  +  hy,  +  g)  +  m{hx^  +  hy^  +  f)  ] 

+  r^al''  +  6m2  -{-2hlm)  =  0 

Denote  these  distances  by  r-^  and  r,.    Then 

1,1-^      2l(ax,  +  Mh  +  d)  +  2mCA.-y,  +  ly,  +  f) 

n        r^  f{x^,  ijO 

Let  the  harmonic  conjugate,  S,  of  (a?i,  y-^)  with  respect  to  the  two 
intersections  of  the  chord  and  the  conic  be  denoted  by  (a?',  i/'), 
then 

PS      V(x'-x,r  +  {y'-yJ 
We  have  also 

^'  —  a?i y'—ih         


Vix  -  x,y  +  {y  -  y,f  y\x  -  x,r  +  (^*  -  y,) ' 

The  necessary  and  sufficient  condition  that  S  may  be  the  harmonic 
conjugate  of  P  with  respect  to  the  two  intersections  of  the  chord 
and  the  conic  is 

r,       r,      PS 
Substitute  the  values  deduced  above  and  reduce  and  we  have,  on 
dropping  accents, 

x{ax^  +  hy^  +  9)  +  ^j{^(f^x  +  ^Vx  +f)+g^i  +  fyi  +  c  =  0 
which  may  also  be  written  in  the  form 

x,{ax  +  hy  +  g)  +  yA^^  +  'by  +  f)  +  9^  +  fy  +  c  =  0. 


Analytic  Geometry.  117 

Either  of  these  is  therefore  the  equation  of  the  polar  of  [co^,  i/J 
with  respect  to  the  conic  f{Xyy)=0. 

The  majority  of  the  properties  already  established  for  poles  and 
polars  do  not  depend  at  all  upon  the  choice  of  the  system  of 
reference,  and  therefore  the  demonstrations  already  given  hold 
also  for  the  case  of  the  general  conic.  If  the  student  is  not  satisfied 
as  to  the  generality  of  any  particular  theorem,  he  should  investi- 
gate the  subject  by  the  aid  of  the  general  equation  just  developed. 

88.  Articles  61  to  63  enable  us  to  find  the 
LENGTH  OF  AXES,    equations  of  the  axes  of  symmetry  of  any 

conic.  This  done,  in  any  particular  case  it  is 
theoretically  an  easy  matter  to  find  the  distances  between  the 
intersections  of  these  axes  with  the  conic,  i.  e.,  the 
lengths  of  the  axes.  Practical  difficulties  of  computation 
are  apt  to  arise  on  account  of  the  frequent  presence  of 
irrationals  in  the  equations.  These  difficulties  may  be  mini- 
mized by  replacing  the  irrational  by  its  decimal  value  carried  to 
such  a  degree  of  approximation  as  the  particular  investigation 
may  demand.  It  is  also  well  to  remember  that  what  is  needed  is 
not  the  co-ordinates  x^,  y^,  so^.  y^  of  the  intersections,  but  the 
quantities  {x^  —  x^Y  and  {y^  —  2/2)^;  and  that  if  a  and  /?  are  the 

roots  of 

ax''  +  2?>a?  +  c  =  0, 

a 

89.  When  the  location  of  the  center  and  the 
FOCI  AND  lengths  and  directions  of  the  axes  of  a  conic 
ECCENTRICITY.           have  been  determined,  the  location  of  the  foci 

and  the  directrices  and  the  determination  of 
the  eccentricity  immediately  follow."^ 

90-  It  has  already  been  shown  that  the  asymp- 

ASYMPTOTES.  totes  of  any  central  conic 

ax'  +  Pif  =  1 

*An  interesting  treatment  of  this  problem  is  given  in  C.  Smith's  Conic 
Sections,  Article    171. 

tFor  a  proof  of  the  existence  of  the  four  foci  different  from  the  one  given 
in  Article  66,  see  C.  Smith,  Conic  Sections,  Article  190. 


lis  Analytic  Geometry. 

referred  to  its  axes  of  symmetry  as  axes  of  co-ordinates,  are  given 
by  the  two  factors  of 

aur  +  liy  =  0 
If  now  we  apply  any  transformation  or  series  of  transformations, 
of  the  sort  hitherto  considered,  and  thus  transform  the  equation 
of  the  conic  to  the  form 

I         ax-  +  hy-  4-  21ixy  +  2gx  +  2f  1/  +  c  =  0 
the  equation  of  the  pair  of  asymptotes  will  become 
II         ax'-  +  &I/2  +  2nxy  +  2gx  -\-2fy  +  k  =  0, 

i.  e.,  the  equation  of  the  pair  of  asymptotes  of  any  central  conic 
differs  from  the  equation  of  the  conic  only  in  the  constant  term. 
(At  first  glance  it  seems  as  if  c  and  k  are  connected  by  the  re- 
lation, c=k — 1,  but  a  moment's  consideration  will  show  that  con- 
stant factors  ma}^  have  been  introduced  at  any  point  in  the  trans- 
formation and  that  in  consequence  c  and  k  may  differ  by  any 
constant.)  The  determination  of  k  presents,  however,  no  serious 
difiaculty.  Equation  II  is  an  equation  with  one  arbitrary  para- 
meter, k,  and  therefore  represents  a  family  of  conies  distinguished 
from  each  other  by  the  various  values  of  k.  In  this  family  the 
pair  of  asymptotes  is  included,  and  the  value  of  k  corresponding 
to  this  particular  member  of  the  family  may  be  determined  by 
giving  algebraic  expression  to  any  one  of  its  additional  properties, 
just  as  in  article  43  we  determined  a  particular  member  of  a 
family  of  lines  by  giving  algebraic  expression  to  some  property  of 
the  line  other  than  the  one  which  it  shared  with  all  the  members 
of  the  family.  Now  the  conic  whose  equation  we  are  seeking,  the 
pair  of  asymptotes,  is  degenerate  and  this  propert^^  of  degeneracy 
is  certainly  not  shared  by  all  the  conies  of  the  family.  If  therefore 
we  impose  on  the  equation  II  the  condition  of  degeneracy,  we  shall 
get  the  values  of  k  corresponding  to  the  degenerate  members  of 
the  family;  included  among  these  will  be  the  value  of  k  corre- 
ciponding  to  the  pair  of  asymptotes.  But  the  condition  of  degen- 
eracy is 

a        h        g 

h        h        /   =0 
9        f        /^ 
an  equation  of  the  first  degree  in  k.    There  is  therefore  only  one 


Analytic  Geometry.  119 

degenerate  member  of  the  family;*  and  the  value  of  k  thus  deter- 
mined, gives  us,  when  substituted  in  equation  II,  the  equation  of 
the  pair  of  asymptotes  of  the  conic  represented  by  equation  I. 

^"l-  If  the  conic  represented  by  our  general 

SPECIAL  equation  of  the  second  degree  is  in  any  par- 

TREATMENT  FOR  ticular  case  a  parabola,  certain  steps  in  our 
THE  PARABOLA.  general  treatment  (such  as  the  determination 
of  the  center  and  the  lengths  of  the  axes) 
become  impracticable,  since  they  introduce  infinite  quantities  into 
the  discussion.  What  we  actually  need  in  order  to  determine  the 
nature  of  a  parabola  from  its  equation  is  the  location  of  its  axis 
of  symmetry  and  the  tangent  at  its  vertex,  and  the  value  of  its 
parameter.  The  discussion  to  follow  makes  use  of  the  following 
theorems : 

(A)  The  semi-parameter  of  a  parabola  is  equal  to  the  distance 
of  any  diameter  from  the  axis  of  symmetry  of  the  parabola,  multi- 
plied by  the  tangent  of  the  angle  which  the  chords  bisected  by  that 
diameter  make  with  the  axis  of  symmetry.  (This  theorem  is  a 
mere  generalization  of  the  solution  of  problem  1,  article  78.) 

(B)  Let 

^1  =  0  and  /Sf2  =  0 

be  any  two  straight  lines.    Then 

is  a  conic  passing  through  the  intersection  of  these  two  lines,  with 
^2  =  0  as  its  tangent  at  this  intersection.  For  if  we  substi- 
tute the  value  of  y  derived  by  solving  /Sfs  =  0  in  the  equation 
^i'  +  ^<^^2 '=  0  the  resulting  equation  in  a?  is  a  perfect  square,  i.  e., 
the  line  /S^s  =  0  meets  the  conic  ^i^  +  1<^'^2  =  0  in  two  coincident 
points  (is  tangent),  and  since,  as  is  easily  seen,  the  co-ordinates  of 
these  points  satisfy  both  >Sfi  =  0  and  S.j,  =  0  the  point  of  tangency 
is  the  intersection  of  the  two  lines. 

Let  it  be  granted  that  the  general  equation 

ax^  +  lif  +  ^.hxij  +  2^^  +  2fi/  +  c  =  0 
represents  a  parabola.    Then  we  have 
ab  —  n^  =  0, 

*Tliat  is,  only  one  with  a  finite  value  of  Ic.  But  the  asymptotes  of  a 
central  conic  are  a  pair  of  lines  in  the  finite  part  of  the  plane  and  hence 
all  the  co-efficients  in  their  equation  are  finite. 


120  Analytic  Geometry. 

and  the  general  equation  of  a  diameter  takes  the  form 

ax  +  \/ah  y  +  ff  +  T  (.Vab  x  +.  hy  +  /)  =  0 

that  is        Ca  +  y\/Mx  +  C;/^  +  t^)  2/  +  ^  +  T  ^=0 


(a  result  which  gives  us  the  theorem  that  all  diameters  of  a  para- 
bola are  parallel) .    For  the  axes  of  symmetry  we  must  have 


(!) 


-S-'(T'-'=» 


i   e  — 


=-^/^'^/! 


Substituting  the  first  of  these  values  in  the  general  equation  ol 
a  diameter  we  have  a  constant  equal  to  zero,  i.  e.,  one  of  the 
diameters  of  the  parabola  is  the  line  at  infinity,  a  fact  which  we 
already  knew.    Substituting  the  other  value  we  have 

(a+h) ^fax  +  («+&)  V%  +  Va;<7  +  V&f  =  0 

i.  e.,  Vax  +  Vhy  +  V(ML±^_=  q 

a  -\-h 

the  equation  of  the  axis  of  symmetry  of  the  parabola.  Since  all 
diameters  of  the  parabola  are  parallel,  the  particular  one  which 
passes  through  the  origin  is 

and  the  perpendicular  distance  from  this  diameter  to  the  axis  of 
symmetry  is 

(a  +  if 
The  original  equation  may  be  put  in  the  form 

( V«^  +  V&2/) '  +  2^0?  +  2f7/ +  c  =  0 
Therefore         "Igx  +  2fi/  +  c  =  0 

is  the  tangent  to  the  parabola  at  the  point  where  it  is  met  by 
the  diameter  through  the  origin.  The  slope  of  this  tangent  and 
therefore  of  the  chords  bisected  by  the  diameter 

■\J~ax-\-  V62/  =  0 

is  — A  and  the  tangent  of  the  angle  which  these  chords  make 


Analytic  Geometry. 
with  tlio  axis  of  symmetry  is 


121 


therefore 


p  = 


V  af —  V  bo 
VcbQ  +  Vhf 
Vaf—yig 

{a  +  if 

The  student  has  now  suflScient  material  at  hand  to  determine 
all  the  data  concerning  any  particular  parabola. 
Example.    As  an  illustration  consider  the  equation 

4a?2  -f  4^2/  +  ?/'  +  6a?  +  2i/  +  4  =  0, 

i.  e.,  (2a?  +  i/)2  +  6^  +  22/  +  4  =  0. 

The  axis  of  symmetry  is 

5 
and  the  semi-parameter  is  ±:  —  ,/5 

The  vertex  of  the  conic  is  its  intersection  with  the  axis  of  sym- 
metry, i.  e., 

79 ,  44^ 

50 '  2^ 


The  tangent  at  the  vertex  is  therefore 

/        44\       1  /     ,    79\ 
(^-25)=2("+50J 

and  the  directrix  is  parallel  to  this 


at  a  distance  of  —  w  5 
50 

All  that  is  needed  to  complete  our 

information  is  to  know  which  way 

the    parabola    is    turned.      This    is 

settled  at  once  by  the  co-ordinates 

of  any  other  point  on  the  conic.    For 

example   the  diameter 

2a?  +  i/  =  0 

meets  the  curve  at  ( — 2,  4)  which 
lies  to  the  left  of  the  tangent  at  the 
vertex.     The   curve  is  therefore   as  drawn. 


Fig.  32. 


122  Analytic  Geometry, 

92.  The  student  has  now  at  his  disposal  sufli- 
HIGHER  LOCI.             cient  formulae  to  enable  him  to  determine 

the  character,  form,  and  location  of  the  curve 
represented  by  any  algebraic  equation  of  the  second  degree.  Loci 
corresponding  to  higher  degree  algebraic  equations  or  to  trans- 
cendental equations,  as  well  as  curves  and  surfaces  in  space,  can 
be  treated  in  a  more  satisfactory  way  after  the  student  has  ac- 
quired a  knowledge  of  the  elements  of  the  differential  and  integral 
calculus. 

93.  PROBLEMS. 

FAMILIES  OF  CONICS. 

1.  Given  that 
^^  =  0        and        ^2  =  0 
are  the  equations  of  two  conies,  show  that 

S^  +  kS^  =  0, 

where  k  is  an  arbitrary  constant,  represents  a  family  of  conies, 
singly  infinite  in  number,  each  of  which  passes  through  all  four 
of  the  intersections  of  the  two  original  conies. 

2.  Five  points  on  a  conic  are  sufficient  to  determine  it  uniquely. 
Therefore  among  the  conies  through  four  given  points,  one  and 
only  one  passes  through  each  of  the  remaining  points  of  the  plane. 
Hence  show  that  the  family 

S^-\-kS2  =  0 

includes  every  possible  conic  through  the  intersections  of 

S,  =  0        and        S,  =  0. 

3.  Given  the  two  conies 

3a?2  -f.  2?y-  4-  4a?  —  1  =  0 
^'  — ^'  +  2t/  — 3  =  0 

form  the  equation  of  the  family  of  conies  through  the  intersections 
of  these  two  conies,  and  find  the  equation  of  that  member  of  the 
family  which  passes  through  the  origin. 

4.  Show  that  the  family  of  the  last  problem  includes  one  circle, 
two  parabolas,  and  three  degenerate  members.  To  what  extent 
are  these  statements  true  of  the  general  case  of  problem  1  ? 

5.  How  many  members  of  the  family  in  problem  1  are  tangent 
to  any  given  line? 


CHAPTER  XVII. 

OTHER  SYSTEMS  OF  CO-ORDINATES, 
POLAR  CO-ORDINATES. 

^^-  We  must  not  assume  that  the  system  of  co- 

various  SYSTEMS    ordinates  we  have  been  using  is  the  only 
OF  CO-ORDINATES,    system  in  use.     On  the  contrary  any  set  of 
quantities  which  will  serve  to  determine  the 
position  of  a  point  may  be  taken  as  co-ordinates  of  the  point. 

In  the  Cartesian  system  the  co-ordinates  of  a  point  are  the 
distances  of  two  lines,  x=^a,  y^^h,  from  the  base  lines  which 
form  the  system  of  reference.  In  other  words  the  point  is  located 
as  the  intersection  of  two  lines. 

Next  in  point  of  simplicity  comes  a  system  in  which  the  point 
is  located  as  the  intersection  of  a  straight  line  and  a  circle.     In 
this  the  system  of  reference  is  a  fixed 
point  A  and  a  fixed  line  AB.  Any  point 
P  in  the  plane  may  now  be  located  by 
giving  p    the  radius  of  the  circle  cen- 
tered at  A  and  passing  through  P,  and 
0    the  angle  between  the  base  line  AB 
and  the  line  AP.     It  is  evident  that  in  ?' 
this  system  a  pair  of  co-ordinates,    p,  Q^ 
determines  the  point  P  not  uniquely, 
but  as  one  of  two,  i.  e.,  either  P  or  P'. 

Next  in  order  of  simplicity  comes  a  system  of  bi-polar  co-ordi- 
nates in  which  the  point  is  located  as 
the  intersection  of  two  circles.  The 
system  of  reference  consists  of  two 
fixed  points  A  and  B  and  the  two  co- 
ordinates are  the  radii  of  two  circles 
centered  at  A  and  B.  It  is  evident  that 
in  this  system  a  pair  of  co-ordinates 
does  not  determine  a  point  uniquely, 
but  as  one  of  two,  i.  e.,  either  P  or  P'. 


124  Analytic  Geometry. 

Consider  still  another  system.  Take 
two  ellipses,  concentric  and  co-axial, 
and  let  the  semi-axes  of  the  one  be  a 
and  2a,  and  the  semi-axes  of  the  other 
2&  and  6.  Then  a  and  6  are  the  co- 
ordinates of  the  intersections  of  the  two 
ellipses.  It  is  evident  that  any  pair  of 
co-ordinates  does  not  determine  a  point 
uniquely,  but  as  one  of  four,  P,  P',  P", 
P'". 

In  general  given  any  two  families  of  curves,  each  of  which 
depends  upon  a  single  arbitrary  parameter,  and  given  also  a  point 
P,  we  can  determine  the  values  of  the  parameters  giving  in 
each  family  the  particular  member  which  passes  through  P.  This 
pair  of  values  may  be  regarded  as  the  co-ordinates  of  P,  deter- 
mining P,  not  uniquely,  but  as  one  of  the  intersections  of  the  two 
curves  given  hj  the  chosen  values  of  the  parameters.  It  is  now 
evident  that  the  number  of  systems  of  co-ordinates  is  infinite. 

^5*  Certain  systems   of  co-ordinates  possess 

MERITS  AND  particular  merit  for  the  investigation  of  some 

DEMERITS  OF  particular  problem.     In  the  bipolar  for  ex- 

VARious  SYSTEMS,  ample,  if  we  take  the  base  points  as  the  foci 
of  an  ellipse  or  hyperbola  the  equations  of 
these  curves  reduce  to  the  simple  forms  x  -{-  y  =  k,  and  x  —  y  =  k. 
On  the  other  hand  each  system  has  certain  demerits.  In  the  bi- 
polar system  for  example,  the  simplicity  of  the  form  taken  by  the 
equations  of  certain  conies  is  more  than  offset  by  the  complexity 
of  the  equation  of  the  straight  line. 

In  one  particular  all  of  the  systems  so  far  discussed  are  seriously 
lacking.  Given  two  algebraic  variables  of  the  most  general  type, 
x  =  Xi  -\-iX2,  and  y  =  yx-\-  iy^^  we  can  assign  to  x^,  x^,  t/i,  2/2  any 
values  whatever,  and  therefore  can  form  a  quadruply  infinite 
number  of  pairs  of  values  of  x  and  y.  But  all  these  systems  of 
co-ordinates  attempt  to  represent  pairs  of  values  of  x  and  y  by 
points  in  the  plane,  and  such  points  are  only  doubly  infinite  in 
number.  In  consequence  each  system  must  fail  to  give  a  complete 
geometric  representation  to  the  algebraic  relations  under  con- 
sideration. Some  of  the  systems,  as  may  be  seen  from  the  prob- 
lems which  follow  this  article,  leave  pairs  of  real  values  of  the 


Analytic  Geometry.  125 

variables  without  point  representation,  while  others  represent  real 
points  by  pairs  of  imaginary  values  of  the  variables. 

The  majority  of  the  systems  of  co-ordinates  which  may  be  used 
fail  also  in  another  important  particular  in  that  they  do  not 
establish  a  one  to  one  correspondence  between  points  and  pairs 
of  values,  but  determine  two  or  more  points  as  corresponding  to  a 
single  pair  of  values.  One  of  the  great  beauties  of  the  Cartesian 
system  is  that  it  establishes  a  one  to  one  correspondence  between 
the  points  of  the  plane  and  all  pairs  of  real  values  of  two  algebraic 
variables. 

PROBLEMS. 

1.  Given  a  system  of  bipolar  co-ordinates  in  which  the  distance 
between  the  base  points  is  8,  plot  the  locus  x  —  y=^0. 

2.  Is  the  point  (3,  3)  on  the  above  locus?  locate  it  in  the 
diagram. 

3.  Plot  the  ellipse  a?  +  7/  ^f=  10  in  the  same  system. 

4.  Plot  the  hyperbola  x  —  7/  =  10  in  the  same  system. 

5.  What  conditions  must  be  met  by  the  co-ordinates  of  a  point 
in  order  that  it  may  be  represented  on  the  diagram  if  the  distance 
between  the  base  points  is  /c  ? 

6.  Given  the  two  families  of  ellipses  whose  equations  in  Carte- 
sian co-ordinates  are 

^^  +  4  =  1.      and      ^-^y'  =  l 


h' 


then  the  values  of  a  and  &  determining  the  ellipses  through  any 
point  are  the  co-ordinates  of  that  point  in  the  new  system  in  which 
the  ellipses  are  the  determining  elements.  What  are  the  Cartesian 
co-ordinates  of  the  points  whose  co-ordinates  in  the  new  system 

are  (4,2),  (4,1),  (1,1),  (1,2)? 

7.  Plot  in  this  same  system  the  locus  for  which  a=h. 

8.  What  are  the  co-ordinates  in  this  system  of  the  point  whose 
Cartesian  co-ordinates  are  (3,  3)  ? 

96.  Polar  co-ordinates  are  of  particular  value 

POLAR  lj^  ^jj^,  investigation  in  which  the  important 

CO-ORDINATES.  ,         '^   .  xi         -,•    .  ^    ^-       ^-  v 

elements  are  the  distance  and  direction  of 
the  variable  point  from  a  fixed  point.     The 

movement  of  the  earth  about  the  sun  or  any  problem  concerning  a 

spiral  curve  are  illustrations. 


126  Analytic  Geometry. 

The  drawbacks  to  the  system  as  it  was  outlined  in  article  94  are 
numerous.  Given  a  pair  of  values  p,  ^,  we  note  first  that  while  0 
may  have  any  value  p  must  be  positive  and  that  the  point  is  deter- 
mined only  as  one  of  two.  On  the  other  hand  given  a  point  ? 
we  have  for  it  a  single  value  of  p,  but  any  number  of  values  of  0 
i.e.,0,0+27r,0-h47T,0+Q7r....0+2mT.  (Hereafter  we  shall  call  thep 
of  any  point  the  radius  vector  and  the  0  the  amplitude  of  the 
point.)  In  other  words  any  real  amplitude  may  be  paired  with  any 
real  positive  radius  vector  and  the  combination  will  be  represented 
by  either  one  of  two  points,  while  a  point  is  represented  by  a  real 
radius  vector  paired  with  an^^  one  of  an  infinity  of  amplitudes 
difi:'ering  from  each  other  by  integer  multiples  of  27r. 

Mathematicians  are  accustomed  however  to  make  an  assumption 
which  enables  them  to  include  negative  values  of  the  radius  vector. 
Abandoning  the  circle  they  fix  their  attention  on  the  angle  0  and 
the  distance p,  and  agree  that  the  positive  value  of  p  is  to  be  meas- 
ured from  the  vertex  of  the  angle  in  the  direction  of  the  boundary 
of  0  and  a  negative  value  of  p  in  the  opposite  direction.  With  this 
agreement  the  co-ordinates  (p,  0)  denote 
the  point  P  and  the  co-ordinates  (  — p,  0) 
the  point  P'.  But  the  point  P  has  now  the 
co-ordinates  (p,  0)  or  (— p,  0  +  it),  or 
more  generally  (p,  0  +  2mr)  or  (— p, 
0  +  {2n  -f  1)77).  In  other  words  any  real 
pair  of  values  of  p  and  0  is  represented 
by    a    single  point  while  any  point  has 

two  radii  vectores  and  an  infinity  of  amplitudes  differing  from 
each  other  by  integer  multiplies  of  tt. 

PROBLEMS. 

Plot  the  curves  corresponding  to  the  following  equations : 

1.  P  =  4.  2.  0=-2. 

3.  p=  0.  4.  p=  sin  0. 

5.  P  =  e^L  ().  0'  -r  W  -r2  =  0. 

7.  Show  that  the  area  of  a  triangle  with  one  vertex  at  the  pole 

(the  fixed  point)  and  the  others  at  (p,,  ^,)  and  (p„  0.^  is  -z  P1P2  sin 

{0,-0.d.  \        ^^ 

8.  ShoAV  that  the  area  of  the  triangle  whose  vertices  are  (p,,    ^1), 

(p.,  0^.  (p..  ft)  is  \{p^P^  sin  (ft  —  ft)  +  P,p,  sin  {0,  —  ft) 

-f  P:iPi  sin  (ft  —  ft)  j- 


Analytic  Geometry. 


127 


97. 


THE    RELATION    OF 
POLAR  AND 
CARTESIAN 
CO-ORDINATES. 

cides  with  the  X  axis, 
have  at  once 


Cartesian  and  polar  co-ordinates  are  con- 
nected by  simple  relations  which  make  the 
transformation  from  one  to  the  other  an  easy 
matter.  We  consider  first  the  case  where  the 
pole  is  at  the  origin  and  the  base  line  coin- 
In  this  case  we 

y 


x  =  p  cos  0 
y  =  p  sin  0 


whence 


X 

When  the  pole  is  at  the  origin  and 
the  base  line  OA  makes  an  angle  a 
with  the  axis  of  X  we  have 

x=  p  cos  {0  -{-  a) 
y=P  sin  (^  +  a) 
whence 


Fig.  37. 


6=  tan-^  '^-  —  a 

^  Fig.  38. 

When  the  Cartesian  system  is  not 
in  either  one  of  these  positions  it  can  easily  be  placed  in  one  or 
the  other  by  a  movement  of  the  axes  parallel  to  themselves.    There- 
fore in  the  most  general  case  when  the  pole  is  at  the  point  {a^l)) 
and   the    base    line   O'A   makes   an 
angle   a  with  the  axis  of  X  we  have 

x  —  a  ==  p  cos  {0  +  a) 
y —  h  =  p  sin  (0  +  a) 
whence 

P  =  V\x  —  cif  +  (?/  —  hj 

0  =  t^n^^rzd>-a 

X  —  a 

If  it  is  desired  to  pass  from  one  fig.  so. 

polar  system  to  another  whose  pole 
is  not  coincident  with  the  first  it  is  simpler  to  transform  first  to  a 


128 


Analytic  Geometry. 


Cartesian  system  whose  origin  and  X  axis  coincide  with  the  pole 
and  base  line  of  the  first  system  and  then  transform  to  the 
second  polar  system. 

PROBLEMS. 


1.  Show  that  the  equation  of  the  circle  centered  at  (/a^ ,  Oi)  and 
of  radius  r  is  p'  —  2pp,  cos  (0  —  60  +  Pi"  ^  /  =  0. 

2.  Find  the  equation  of  a  circle  when  the  pole  is  on  the  circum- 
ference and  the  base  line  is  the  tangent  at  the  pole. 

8.  Find  the  equations  of  the  various  conic  sections  when  the 
pole  is  at  one  focus  and  the  base  line  is  the  axis  of  symmetry 
through  that  focus,  and  show  that  any  one  of  them  may  be  reduced 
to  the  form 


^=1 


e' cos  0 


where  p  and  e  are  the  semi-parameter 
and  the  eccentricity. 

4.  Let  AB  be  the  directrix,  F  the 
focus,  and  P  any  point  on  the  conic. 
Deduce  the  equation  just  given  directly 
from  the  diagram.  (It  is  sometimes 
convenient  to  measure  the  angle  0  from 
l^C  as  a  base  line.  Li  that  case  the 
equation  takes  the  form 

—  =  l  +  e  cos  0). 
P 


■ 

c 

3 

f/T 

A 

F 

^ 

Fig.  40. 


APPENDIX  A. 

INFINITIES  OF  VARIOUS  ORDEES. 

The  student  must  bear  in  mind  the  fact  that  he  is  using  the 
word  infinity  in  a  technical  sense  differing  somewhat  from  the 
ordinary  literary  and  philosophical  usage  of  the  word.  In  its 
ordinary  usage  infinity  denotes  that  which  exceeds  all  limitations 
and  therefore  no  comparisons  between  various  infinities  are  pos- 
sible. In  mathematical  usage  infinity  denotes  that  which  increases 
indefinitely,  and  it  is  evident  that  between  two  such  quantities  a 
perfectly  definite  comparison  may  be  made.  Consider  some  illus- 
trations for  the  sake  of  clearness.    In  each  of  the  three  expressions 

2,y  +  1         x'  +  1        x'  +  1 
X  X  x" 

let  X  increase  indefinitely.  As  this  happens  both  numerator  and 
denominator  of  each  fraction  increase  indefinitely,  i.  e.,  in  tech- 
nical phrase,  become  infinite,  but  the  three  fractions  respectively 
tend  to  2,  increase  indefinitely  (tend  to  infinity),  and  tend  to  zero. 
In  other  words,  in  the  first  case  while  both  numerator  and  denom- 
inator increase  they  remain  easily  comparable  with  each  other;  in 
the  second  case  the  numerator  becomes  incomparably  greater  than 
the  denominator,  and  in  the  third  case  the  numerator  becomes  in- 
comparably less  than  the  denominator.  Mathematicians  express 
all  this  by  saying  that  in  the  first  case  numerator  and  denominator 
are  infinities  of  the  same  order,  that  in  the  second  case  the  num- 
erator is  an  infinity  of  higher  order  than  the  denominator,  that  in 
the  third  case  the  numerator  is  an  infinity  of  lower  order  than  the 
denominator.  In  general  given  any  two  infinities  x  and  y,  x  is  said 
to  be  of  the  same  order,  a  higher  order,  or  a  lower  order  than  y 

according  as  the  ratio  —  tends  to  a  finite  quantity,  infinity,  or 

zero.  If  it  is  desired  to  make  a  still  more  accurate  distinction 
some  one  infinity  (in  general  the  one  of  lowest  order  among  those 
under  consideration)  is  called  an  infinity  of  the  first  order  and  its 


130  Analytic  Geometry. 

square,  cube,  ith  power  called  infinities  of  the  second,  third,  ith 
order.  The  order  of  any  other  infinity  is  then  determined  by 
comparison  with  these  powers  of  the  chosen  infinity.  Stated  alge- 
braicly,  let  z  be  an  infinity  of  the  first  order  and  y  any  other 
infinity.    Then  y  is  said  to  be  an  infinity  of  the  A;th  order  when 

the  limit  of  ^  as  1/  and  z  tend  to  infinity  is  a  finite  quantity  differ- 
ent  from  zero. 


APPENDIX  B. 
FUNCTIONALITY. 

One  variable  is  said  to  be  a  function  of  another  when  the  two  are 
so  related  that  a  change  in  the  one  produces  a  change  in  the  other.* 
Thus  the  momentum  of  a  moving  body  is  a  function  both  of  its 
mass  and  of  its  velocity.  Many  other  examples  of  functionality 
in  nature  may  easily  be  given. 

Any  equation  between  two  variables  defines  either  as  a  function 
of  the  other.    For  example 

x^  +  y^  =  4. 

so  connects  x  and  y  that  no  change  can  be  made  in  either  without 
affecting  the  other.  Consequently  whether  we  shall  call  x  a  func- 
tion of  7/  or  2/  a  function  of  a?  is  a  matter  of  purely  arbitrary  choice. 
If  the  equation  be  solved  for  either  of  the  variables,  the  one  for 
which  it  is  solved  is  said  to  be  an  explicit  function  of  the  other. 
Otherwise  it  is  called  an  implicit  function.  Thus  in  the  example 
last  given  x  is  an  implicit  function  of  y,  but  solve  the  equation  for 
X  arid  we  have 


which  defin^es  x  as  an  explicit  function  of  y. 

AVhen  for  any  reason  it  becomes  desirable  to  indicate  the  fact 
of  functional  relation  without  specifying  its  exact  nature  we  use 
the  form 

y=f(x) 

which  is  variously  read  y  equals  a  function  of  x,  y  equals  the  f 
function  of  x,  or  more  simply  y  equals  the  f  of  x.  When  several 
functions  enter  into  the  discussion  they  are  distinguished  by  the 
use  of  subscripts  fi,  f^,  f^,  etc.,  or  by  the  use  of  other  letters  F^  cf> 


*This  definition  of  functionality  is  merely  a  Avorking  definition  for  present 
use.  In  some  branches  of  the  higher  mathematics  it  becomes  necessary  to 
define  more  closely  and  to  recognize  distinctions  which  the  student  is  not 
now  prepared  to  consider. 


132  Analytic  Geometry. 

i}/  etc.,  in  the  place  of  f.  When  in  any  discussion  the  form  of  the 
function  corresponding  to  any  functional  symbol,  F{x)  for  ex- 
ample, has  been  defined,  it  is  understood  that  this  symbol  shall 
continue  to  represent  this  same  form  throughout  the  discussion. 
For  example  if  we  have 

f{w)=a}''  +  2x  —  a 
then 

f(y)=y'+^y—a 

f{c)=c^  -\-2c  —  a 


APPENDIX  C. 
PERMISSIBLE  OPERATIONS. 

In  reducing  equations  to  a  simple  form  as  in  article  24,  the 
student  must  see  to  it  that  the  operations  performed  do  not  in  an}^ 
way  change  the  character  of  the  locus.  The  force  of  this  remark 
is  most  clearly  set  forth  by  some  illustrations. 

The  equations 

2a!  -\-  y  =  x  —  4 
and  X  -{-y  -\-  4:=0 

are  equivalent,  i.  e.,  represent  the  same  locus,  since  every  point 
which  satisfies  either  one  satisfies  the  other  also. 
The  equations 

x4-y—-^  =  0 

and  3^  +  3i/— 7  =  0 

are  equivalent  since  every  point  which  satisfies  either  satisfies 
the  other  also. 
The  equations 

x^  -\-  xy  —  2a?  =  0 
and  X -\- y  —  2  =  0 

are  not  equivalent  since  the  first  is  satisfied  by  every  point  on  the  Y 
axis  and  the  second  is  not. 
The  equations 

x  =  y 
and  x^  =  y^  » 

are  not  equivalent  since  the  latter  is  satisfied  by  points  for  which 
x  =  —  1/  as  well  as  by  those  for  which  x  =  y. 

A  little  consideration  will  show  that  the  modifications  to  which 
we  subject  our  equations  may  be  reduced  to  two  operations :  trans- 
position, and  the  introduction  and  rejection  of  factors.  Stili 
further  consideration  will  show  that  a  transposition  of  terms  can 


134  Analytic  Geometry. 

in  no  way  affect  the  locus,  and  that  the  introduction  or  rejection 
of  a  factor  has  no  effect  upon  the  locus,  provided  that  the  factor 
is  of  such  a  nature  that  it  cannot  vanish  for  any  values  of  the 
variables.  The  student  will  do  well  at  this  time  to  read  the  articles 
in  some  standard  algebra  upon  reversible  operations.  See  in 
particular  Fine's  College  Algebra. 


APPENDIX  D. 
PROJECTION. 

Let  P  be  any  point  and  CD  any  plane. 
Let  Q  be  any  other  point.  Draw  PQ 
and  let  R  be  the  point  in  which  PQ  ex- 
tended meets  CD.  Then  R  is  the  pro- 
jection of  Q  on  the  plane  CD  from  P, 
sometimes  called  the  center  of  projec- 
tion. The  projection  of  any  geometric 
object  or  group  of  objects  is  the  aggre- 
gate of  the  projections  of  all  the  points     ^"^ ^^^  ^^ 

of  the  object  or  group  of  objects.  •  The 

projection  is  sometimes  spoken  of  as  the  shadow  of  the  object  cast 
on  the  plane  CD  by  a  light  at  P,  and  this  description  is  satis- 
factory when  the  point  P  and  the  plane  CD  are  on  opposite  sides 
of  the  object.  When  this  is  not  the  case,  as  for  Q'  above,  it  is 
necessary  to  fall  back  on  the  definition  first  given.  If  this  point  P 
removes  indefinitely,  the  lines  joining  P  to  the  various  points  of 
the  object  to  be  projected  tend  to  parallelism,  i.  e.,  projection 
from  an  infinite"  distance  is  by  parallel  lines.  If  in  addition  P 
removes  indefinitely  in  a  direction  perpendicular  to  the  plane  CD, 
the  projecting  lines  tend  to  parallelism  and  perpendicularity  to 
CDj  i.  e.,  the  perpendicular  projection  from  an  infinite  distance 
consists  of  the  aggregate  of  the  feet  of  the  perpendiculars  let  fall 
from  all  the  points  of  the  object  upon  the  plane  of  projection. 
In  this  case  the  projection  is  said  to  be  orthogonal  and  is  evidently 
the  sort  already  presented  to  the  student  in  his  study  of  solid 
geometry. 

If  all  the  points  of  the  object  to  be  projected  are  in  one  plane 
EF  and  the  center  of  projection  P  is  in  the  same  plane  it  is  evident 
that  the  projection  will  be  wholly  in  that  plane  and  will  consist 
of  the  totality  of  points  in  which  lines  from  P  to  every  point  of 
the  object  meet  LM,  the  line  of  intersection  of  CD  and  EF.  If  P, 
remaining  in  the  same  plane,  removes  indefinitely  in  a  direction 


136  Analytic  Geometry. 

perpendicular  to  the  line  LM^  the  projection  of  the  object  is  the 
totality  of  the  feet  of  the  perpendiculars  let  fall  from  every  point 
of  the  object  on  LM.  In  this  case  the  projection  is  again  called 
orthogonal,  and  is  evidently  the  sort  already  presented  to  the 
student  in  his  study  of  plane  geometry.  Hereafter  when  no  center 
jP  of  projection  is  mentioned  it  is  assumed  that  the  projection 
is  orthogonal. 

If  the  student  has  a  clear  understanding  of  the  preceding  state- 
ments, the  following  propositions  need  no  proof. 

I.  The  projection  of  any  length  AB  upon  the  straight  line  CD^ 
which  makes  with  AB  an  angle  ^,  is  AB  cos^. 

II.  The  projection  of  any  contour 
is  the  sum  of  the  projections  of  its 
component  parts.  Proj.  FQRS  = 
Proj.  P§  +  Proj.  QR +  Froj.  R8. 

III.  For  purposes  of  projection  a  ^^^'  ^^' 

curved  line  may  be  regarded  as  the  limiting  form  of  a  broken  line 
as  the  number  of  its  component  parts  increases  indefinitely  and 
each  part  tends  to  zero. 

When  the  contour  turns  back  upon  itself  the  question  at  once 
arises  whether  or  not  the  double  portion  shall  be  twice  counted. 
So  far  as  anything  we  have  so  far  indicated  is  concerned  the 
answer  is  yes.  There  is  however  another  point  of  view.  It  is 
evident  that  if  PQ  be  regarded  as  the  path  of  a  moving  point,  the 


Fig.  43. 

projection  of  PQ  on  CD  is  the  amount  of  movement  of  the  point  in 
the  direction  CD  while  the  point  goes  from  P  to  Q.  Looking  at 
the  question  from  this  point  of  view,  which  is  an  important  one  in 
much  mathematical  work,  it  is  evident  that  the  total  movement 
in  the  direction  CD  of  the  point  which  travels  such  a  path  as 
PQRST  is  the  sum  of  the  projections  of  PQ,  QR,  RS,  minus  the 
projection  of  ST.,  i.  e.,  PS'  —  T'S'  =  P'T'. 

The  point  of  view  just  outlined  is  the  one  usually  taken  by 
mathematical  writers,  and  in  accordance  with  it  they  adopt  the 
following  conventions : 


Analytic  Geometry.  137 

I.  In  considering  any  contour  determine  an  initial  and  a  ter- 
minal point. 

II.  The  positive  direction  in  any  portion  of  a  contour  is  the 
direction  of  motion  of  a  point  going  from  the  initial  to  the  ter- 
minal point. 

III.  Assume  one  direction  or  the  other  along  the  line  on  which 
projection  is  made  as  positive. 

IV.  The  projection  of  any  of  the  component  lines  of  a  contour 
is  equal  to  the  length  of  the  line  multiplied  by  the  cosine  of  the 
angle  between  the  positive  direction  of  the  projected  line  and 
the  positive  direction  of  the  line  on  which  the  projection  is  made. 

The  student  may  now  prove  the  following  theorems : 

I.  The  projection  of  any  contour  is  equal  to  the  algebraic  sum 
of  the  projections  of  its  component  parts. 

II.  The  projection  of  any  closed  contour  is  zero. 

III.  The  projections  of  any  two  contours  having  the  same 
initial  and  terminal  points  are  equal. 

IV.  The  resultant  of  any  open  contour  is  defined  to  be  a  line 
joining  its  initial  point  to  its  terminal  point.  Show  that  the  pro- 
jection of  any  open  contour,  is  equal  to  the  projection  of  its 
resultant. 


APPENDIX  E. 
IMAGINARIES. 

lu  all  work  with  complex  nnimbers  the  student  must  be  careful 
to  keep  in  mind  the  fact  that  the  term  imaginary  is  used  in  a 
purely  technical  sense.  In  their  earlier  mathematical  use,  the 
terms  real  and  imaginary  undoubtedly  had  the  significance  they 
now  have  in  ordinary  usage,  but  a  clearer  understanding  of  the 
nature  of  number  has  come  with  the  years  and  we  no  longer  regard 
V  —  1  as  imaginary  in  the  literary  sense  of  the  word  any  more 
than  we  think  of  a  fraction  as  a  broken  number. 

The  whole  matter  will  perhaps  be  clearer  if  we  look  at  the 
nature  of  the  various  sorts  of  number  which  are  sometimes 
grouped  together  as  algebraic.  Algebra  recognizes  six  operations, 
three  direct  (addition,  multiplication,  involution),  three  inverse 
(subtraction,  division,  evolution).  As  material  to  which  to  apply 
these  operations  the  world  about  us  presents  nothing  but  positive 
integers.  There  are  in  nature  no  negative  numbers  and  no  frac- 
tions. An  object  divided  into  two  parts  becomes  two  objects,  and  it 
is  only  by  imagining  an  undivided  object  to  be  divided,  or  imagin- 
ing two  objects  to  be  united  that  we  are  able  to  talk  of  halves. 

If  now  we  apply  any  or  all  of  the  direct  algebraic  operations 
to  the  positive  integers  we  get  nothing  new;  sums,  products,  and 
positive  integer  powers  of  positive  integers  are  all  positive  in- 
tegers, a  fact  which  we  may  express  by  the  statement  that  the 
positive  integers  form  a  complete  group  out  of  which  it  is  impos- 
sible to  pass  by  means  of  the  direct  algebraic  operations.  If 
however  we  apply  to  these  positive  integers  the  inverse  operation 
of  subtraction  we  may  as  before  get  positive  integers,  but  we  get 
also  a  new  sort  of  thing  to  which  we  give  the  name  of  negative 
integers.    The  equations 

5  —  6  =  ^1,        4  —  6  =  ^21        3  —  6  =  ^3,        etc., 

define  a  set  of  numbers  t^,  t^,  t^,  of  such  a  nature  that  when  they 
are  increased  by  1,  2,  3,  they  become  zero,  i.  e.,  they  bear  the  same 


AnxVlytic  Geometry.  139 

relation  to  zero  that  zero  bears  to  1,  2,  3.    It  is  also  evident  that 

t-s  +  ^  =  h,  t,  +  i  =  t,,  t,  +  1=0, 

i.  e.,  ^3,  t^,  t^,  differ  from  each  other  in  the  same  way  as  any  three 
consecutive  integers.  In  other  words  the  inverse  operation  of 
subtraction  enables  us  to  define  a  series  of  negative  integers,  each 
one  symmetric  with  respect  to  zero  to  a  corresponding  positive 
integer,  and  forming  with  zero  and  the  positive  integers  an  un- 
limited sequence  of  numbers  such  that  it  is  possible  to  pass  from 
any  one  to  those  consecutive  with  it  on  either  side  by  the  addition 
or  subtraction  of  unity. 

Now  while  nature  knows  no  such  things  as  a  negative  number 
and  laughs  at  the  idea  of  continuing  the  process  of  subtraction 
after  zero  is  reached,  she  nevertheless  presents  numerous  illus- 
trations of  sequences  of  numbers  arranged  symmetrically  with 
respect  to  some  neutral  point.  Distance  above  and  below  sea  level, 
north  and  south  of  the  equator,  or  east  and  west  of  some  chosen 
meridian ;  temperature  above  and  below  the  freezing  point ;  assets 
and  liabilities,  profit  and  loss  are  examples  of  such  sequences,  any 
one  of  which  may  be  used  to  illustrate  the  properties  and  laws  of 
operation  connected  with  negative  numbers. 

If  we  continue  to  apply  the  inverse  algebraic  operations  we 
derive  other  new  numbers  to  which  we  give  the  names  of  fraction, 
irrational,  imaginary ;  and  then  find  that  with  positive  and  negative 
integers,  fractions,  irrationals,  and  imaginaries  the  field  is  again 
closed,  in  that  no  finite  number  of  algebraic  operations  direct  or 
inverse  can  give  us  anything  new.  For  each  of  these  new  things, 
with  the  exception  of  the  imaginary,  it  is  possible  to  find  with  but 
little  difficulty  excellent  illustrations  in  nature,  and  thereby 
render  clearer  to  our  minds  the  laws  under  which  we  work  with 
these  new  symbols. 

It  is  even  possible  to  find  in  nature  an  illustration  of  imagin- 
aries. Let  us  adopt  the  usual  convention  and  represent  positive 
and  negative  numbers  by  points  on  a  straight  line.    Consider  any 

positive  number  a  and  multiply  it  hji(=^/ — 1)  four  times  in  suc- 
cession, arriving  thus  at  the  numbers 
ia,  —  a,  —  ia,  a.    In  other  words,  two  ' 

multiplications  by  i  have  the  same  effect 

upon  a  as  if  it  had  been  rotated  180  degrees  on  a  circle  centered  at 
0  and  of  radius  a,  while  four  multiplications  are  equivalent  to  a 


140 


Analytic  Geometry. 


,> 


rotation  through  360  degrees.  We  may  therefore,  for  purposes 
of  illustration,  regard  multiplication  by  i  as  equivalent  to  a  rota- 
tion of  90  degrees.  From  this  point  of  view  real  and  pure  imag- 
inary numbers  may  be  represented  by  points  on  two  perpendicular 
lines  as  in  the  diagram.  A  complex  num- 
ber, such  as  a  +  ihy  may  be  represented  by 
a  point  with  an  abscissa  a  and  an  ordinate 
h;  and  thus  by  utilizing  all  the  points  of 
the  plane  we  can  represent  all  values  of  a 
single  complex  variable.  The  assumptions  - 
by  which  this  method  of  representation  is 
built  up  may  seem  arbitrary  or  artificial, 

but  they  amount  merely  to  a  recognition  of  the  direction  as  well 
as  the  distance  of  a  point  from  the  origin,  and  so  differ  in  degree 
rather  than  in  kind  from  the  assumjitions  by  which  we  represented 
positive  and  negative  numbers  by  points  on 
a  straight  line. 

Given  any  complex  number  a  -\-  ib  repre- 
sented by  the  point  P,  the  angle  0  is  called 
the  amplitude  or  argument  of  P;  and  /a, 
always  considered  positive,  is  called  the  mod- 
ulus of  P.    Evidently 


Fig.  45. 


Fig.  46. 


and 


a  =  p  COS  0^       p  =  +  ya   +  b^ 

h=p  sin  0        0  =  tan-'  - 

a 

a  -^  ib  =  p  (cos  0  +  i  sin  0). 


The  term  modulus  is  frequently  abbreviated  to  mod.  or  denoted  by 
a  pair  of  vertical  lines  including  the  quantity  considered.    Thus 


modulus(a+  ih)^mod{a-\-  ib)=  |a  +  i&|  =-\-\/a-  +  &^. 

If  a  complex  number  is  zero  the  real  part  and  the  coefficient  of 
the  imaginary  part  are  both  zero,  and  conversely. 

If  a  complex  number  is  zero  its  modulus  is  zero  and  conversely. 

The  sum  of  two  complex  numbers  consists  of  the  sum  of  the  real 
parts  plus  i  times  the  sum  of  the  coefficients  of  the  imaginary 
parts.  This  theorem  may  be  demonstrated  instantly  by  adding  the 
two  complex  quantities  in  the  form  a  +  i^,,  c  +  id.    Geometrically, 


Analytic  Geometry. 


141 


Fig.  47. 


let  the  point  P  represent  «  +  /&,  and 
Q  represent  c  +  id.  Draw  from  P  a 
line  PR  equal  and  parallel  to  OQ. 
Then  R  represents  a  +  ih  -^c  -\-id  and 

0R=  \a^ib  ^c-\-id\. 
The  justification  of  this  geometric  treat- 
ment of  addition  is  left  to  the  student. 
If  he  has  ever  studied  physics,  the  idea 
will  doubtless  suggest  itself  to  him  that 
complex  quantities  might  with  profit  be 
used  in  the  discussion  of  such  topics  as 
the  resolution  and  composition  of  forces  or  motions. 

The  geometric  treatment  of  addition  leads  at  once  to  the  import- 
ant theorem  that  the  modulus  of  the  sum  of  two  (or  any  number 
of)  complex  numbers  is  less  than,  or  at  most  equal  to,  the  sum  of 
the  moduli.  Is  it  possible  to  state  a  similar  theorem  concerning 
the  amplitudes? 

To  form  the  product  of  Iavo  complex  numbers  a  -]-ih,  c  +  id 

let  \a  +  ih\  =  p,  amp.(«  +  ih)  =  0^  k  +  id\  =  p-,  amp.  (c  +  {d)=  <l>, 
then  (a  +  ib)  (c  +  id)  =  p  (cos  0  +  i  sin  0)  Kcos  <f>  +  i  sin  <^) 

=  PP-  (cos  6  cos  <^  —  sin  $  sin  </>  +  i  (cos  0  sin  <^  +  sin  0  cos  <^)) 

=  PM  (cos  (0  +  4>)  -^  i  sin (0  +  <^)) 
It  follows  at  once  that  the  modulus  of  the  product  of  two  complex 
numbers  is  the  product  of  the  moduli,  and  the  amplitude  of  the 
product  is  the  sum  of  the  amplitudes. 

This  last  theorem  leads  to  a  simple  geometric  method  of  con- 
structing the  point  which  represents  the  product  of  two  complex 
numbers.    Let  P  represent  a-{-il)^  p  (cos  ^  +  i  sin  ^  j  and  Q  repre- 
sent c  -\-  id=  /x  (COS  (^  + 1  sin  <^  )  and 
OA  be  the  unit  of  measure  on  which 
the  diagram  is  constructed.     Construct 
the  angle  if/=^  0  +  4>  and  at  Q  construct 
the  angle  a  =OAP.      Then  R,  the  in- 
tersection  of  the   two   lines   so   deter- 
mined,     represents      .(»  +  i&)  (c  +  id) 
=  /om(c()S   (  0  +  4>)    +   i  sin(  0  +  <A)) 
since  its  amplitude  is  the  sum  of  the 

OP  _    OE , 
OA  OQ 

i.  e.,  OR^=OP.OQ)   its  modulus  is  the  product  of  the  moduli. 


amplitudes  and  (since 


142  AnxVlytic  Geometry. 

The  student  who  desires  to  make  a  further  study  of  this  mode 
of  representing  a  complex  variable  may  read,  among  others, 
Burnside  &  Panton,  Theory  of  Equations,  Vol.  I,  Chap.  XII;  or 
Durege,  Elements  of  the  Theory  of  Functions  of  a  Complex 
Variable,  Introduction. 


FOURTEEN  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 


This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 

Renewed  books  are  subject  to  immediate  recall. 


MAY     4^«»^  ' 


leJan'56Ke 


2Sreb'59Aj 


JAW  4     1956  LU 


aJl 


ar56E0 


^'^CT>  rt5 


MAR  6     1956  lU 


^^&  14,  rggg 


27Apr'5^BH 


APRl3l956Hr 


.^.S^L^ 


^i* 


22Maj'56l.t(f. 


M/>Y'5   -I956(.u 


JUN4.    1057    T 


^:^, 


IV^ 


LD  21-100m-2,'55 
(B139s22)476 


General  Library 

University  of  California 

Berkeley 


^329746 


.) 


